Hodge-theoretic analysis on manifolds with boundary, heatable currents, and Onsager’s conjecture in fluid dynamics

Khang Manh Huynh

Abstract We use Hodge theory and functional analysis to develop a clean approach to heat flows and Onsager’s conjecture on Riemannian manifolds with boundary, where the weak solution lies in the trace-critical 1 3 Besov space B3,1. We also introduce heatable currents as the natural analogue to tempered distributions and justify their importance in Hodge theory.

Acknowledgments The author thanks his advisor Terence Tao for the invaluable guidance and patience as the project quickly outgrew its original scale. The author is also grateful to Jochen Gl¨uck for recommending Stephan Fackler’s insightful PhD thesis [Fac15].

Contents

1 Introduction 2 1.1 Onsager’s conjecture ...... 2 1.2 Modularity ...... 4 1.3 Motivation behind the approach ...... 4 1.4 Blackboxes ...... 5 1.5 For the specialists ...... 6

2 Common notation 7

3 Onsager’s conjecture 8 3.1 Summary of preliminaries ...... 8 3.2 Searching for the proper formulation ...... 12 3.3 Justification of formulation ...... 13

arXiv:1907.05360v2 [math.AP] 14 Jul 2019 3.4 Heating the nonlinear term ...... 14 3.5 Proof of Onsager’s conjecture ...... 15

4 Functional analysis 20 4.1 Common tools ...... 20 4.2 Interpolation theory ...... 20 4.3 Stein extrapolation of analyticity of semigroups ...... 23 4.3.1 Semigroup definitions ...... 24 4.3.2 Simple extrapolation (with core) ...... 25 4.3.3 Coreless version ...... 26

1 4.4 Sectorial operators ...... 27

5 Scalar function spaces 30 5.1 On Rn ...... 30 5.2 On domains ...... 31 5.3 Holder & Zygmund spaces ...... 33 5.4 Interpolation & embedding ...... 34 5.5 Strip decay ...... 35

6 Hodge theory 38 6.1 The setting ...... 38 6.1.1 Vector bundles ...... 38 6.1.2 Compatibility with scalar function spaces ...... 39 6.1.3 Complexification issue ...... 41 6.2 Differential forms & boundary ...... 41 6.3 Boundary conditions and potential theory ...... 44 6.4 Hodge decomposition ...... 47 6.5 An easy mistake ...... 52

7 Heat flow 52 7.1 L2-analyticity ...... 52 7.2 Lp-analyticity ...... 53 7.3 W 1,p-analyticity ...... 55 7.4 Distributions and adjoints ...... 57 7.5 Square root ...... 60 7.6 Some trace-zero results ...... 61

8 Results related to the Euler equation 62 8.1 Hodge-Sobolev spaces ...... 62 8.2 Calculating the pressure ...... 64 8.3 On an interpolation identity ...... 64

A Complexification 67

Nomenclature 69

1 Introduction

1.1 Onsager’s conjecture Recall the incompressible Euler equation in fluid dynamics:

 ∂ V + div (V ⊗ V ) = − grad p in M  t div V = 0 in M (1)  hV, νi = 0 on ∂M

 (M, g) is an oriented, compact smooth Riemannian manifold with smooth boundary, dimension ≥ 2  where ν is the outwards unit normal vector field on ∂M.  I ⊂ R is an open interval, V : I → XM, p : I × M → R.

2 Observe that the Neumann condition hV, νi = 0 means V ∈ XN , where XN is the set of vector fields on M which are tangent to the boundary. Note that when V is not smooth, we need the trace theorem to define the condition (see Subsection 5.2).

Roughly speaking, Onsager’s conjecture says that the energy kV (t, ·)kL2 is a.e. constant in time when V 1 is a weak solution whose regularity is at least 3 . Making that statement precise is part of the challenge. 1 In the boundaryless case, the “positive direction” (conservation when regularity is at least 3 ) has been known for a long time [Eyi94; CET94; Che+08]. The “negative direction” (failure of energy conservation 1 when regularity is less than 3 ) is substantially harder [LS12; LJ14], and was finally settled by Isett in his seminal paper [Ise18] (see the survey in [LS19] for more details and references). Since then more attention has been directed towards the case with boundary, and its effects in the generation of turbulence. In [BT17], the “positive direction” was proven in the case M is a bounded domain n 3 0,α 1 in R and V ∈ Lt C XN (α > 3 ). The result was then improved in various ways [DN18; Bar+18; BTW19]. 3 α 1 In [NN18], the conjecture was proven for V in Lt B3,∞X (α > 3 ) along with some “strip decay” conditions for V and p near the boundary (more details in Subsection 3.2). Most recently, the conjecture was proven as n 3 1/3 part of a more general conservation of entropy law in [Bar+19], where M is a domain in R , V ∈ Lt B3,VMOX 1/3 1/3 (where B3,VMOX is a VMO-type subspace of B3,∞X), along with a “strip decay” condition involving both V and p near the boundary (see Subsection 3.2). Much less is known about the conjecture on general Riemannian manifolds. The key arguments on flat spaces rely on the nice properties of convolution, such as div (T ∗ φε) = div (T ) ∗ φε where T is a tensor field ε↓0 and φε −−→ δ0 is a mollifier, or that mollification is essentially local. This “local approach” by convolution does not generalize well to Riemannian manifolds. In [IO15] – the main inspiration for this paper – Isett and Oh used the heat flow to prove the conjecture on compact Riemannian manifolds without boundary, 1 1 1 for V ∈ L3B 3 X (where B 3 X is the B 3 -closure of compactly supported smooth vector fields). The t 3,c(N) 3,c(N) 3,∞ situation becomes more complicated when the boundary is involved. Most notably, the covariant derivative behaves badly on the boundary (e.g. the second fundamental form), and it is difficult to avoid boundary terms that come from integration by parts. Even applying the heat flow to a distribution might no longer be well-defined. This requires a finer understanding of analysis involving the boundary, as well as the properties of the heat flow. In this paper, we will see how we can resolve these issues, and that the conjecture still holds true with the boundary:

Fact. Assuming M as in Equation (1), conservation of energy is true when (V, p) is a weak solution with 1 3 3 V ∈ Lt B3,1XN . It is not a coincidence that this is also the lowest regularity where the trace theorem holds. We also note a very curious fact that no “strip decay” condition involving p (which is present in different forms for 1 the results on flat spaces) seems to be necessary, and we only need p ∈ Lloc (I × M) (see Subsection 3.3 for details). One way to explain this minor improvement is that the “strip decay” condition involving V naturally originates from the trace theorem (see Subsection 3.3), and is therefore included in the condition 1 3 3 V ∈ Lt B3,1XN , while the presence of p is more of a technical artifact arising from localization (see [Bar+19, Section 4]), which typically does not respect the Leray projection. By using the trace theorem and the heat flow, our approach becomes global in nature, and thus avoids the artifact. Another approach is to formulate the conjecture in terms of Leray weak solutions like in [RRS18], without mentioning p at all, and we justify how this is possible in Subsection 3.3. 1 A more local approach, where we assume V ∈ L3B 3 X as in [IO15], and the “strip decay” condition as t 3,c(N) 1 3 in [Bar+19, Equation 4.9], would be a good topic for another paper. Nevertheless, B3,1XN is an interesting

3 space with its own unique results, which keep the exposition simple and allow the boundary condition to be natural.

1.2 Modularity The paper is intended to be modular: the part dealing with Onsager’s conjecture (Section3) is relatively short, while the rest is to detail the tools for harmonic analysis on manifolds we will need (and more). As we will summarize the tools in Section3, they can be read independently.

1.3 Motivation behind the approach Riemannian manifolds (and their semi-Riemannian counterparts) are among the most important natural settings for modern geometric PDEs and physics, where the objects for analysis are often vector bundles and differential forms. The two fundamental tools for a harmonic analyst – mollification and Littlewood- Paley projection via the Fourier transform – do not straightforwardly carry over to this setting, especially when the boundary is involved. Even in the case of scalar functions on bounded domains in Rn, mollification arguments often need to stay away from the boundary, which can present a problem when the trace is nonzero. Consider, however, the idea of a special kind of Littlewood-Paley projection which preserves the boundary conditions and commutes with important operators such as divergence and the Leray projection, or using the principles of harmonic analysis without translation invariance. It is one among a vast constellation of ideas which have steadily become more popular over the years, with various approaches proposed (and we can not hope to fully recount here). For our discussion, the starting point of interest is perhaps [Str83], in which Strichartz introduced to analysts what had long been known to geometers, the rich setting of complete Riemannian manifolds, where harmonic analysis (and the Riesz transform in particular) can be done via the Laplacian and the heat semigroup et∆, constructed by dissipative operators and Yau’s lemma. Then in [KR06], Klainerman and Rodnianski defined the L2-heat flow by the spectral theorem and used it to get the Littlewood-Paley projection on compact 2-surfaces. In [IO15], Isett and Oh successfully tackled Onsager’s conjecture on Riemannian manifolds without boundary by using Strichartz’s heat flow. These results hint at the central importance of the heat flow for analysis on manifolds. But it is not enough to settle the case with boundary, especially when derivatives are involved. Some pieces of the puzzle are still missing. To paraphrase James Arthur (in his introduction to the trace formula and the Langlands program), there is an intimate link between geometric objects and “spectral” phenomena, much like how the shape of a drum affects its sounds. For a Riemannian manifold, that link is better known as the Laplacian – the generator of the heat flow – and Hodge theory is the study of how the Laplacian governs the cohomology of a Riemannian manifold. An oversimplified description of Fourier analysis on Rn would be “the spectral theory of the Laplacian” [Str89], where the heat kernel is the Gaussian function, invariant under the Fourier transform and a possible choice of mollifier. Additionally, the Helmholtz decomposition, originally discovered in a hydrodynamic context, turned out to be a part of Hodge theory. It should therefore be no surprise that Hodge theory is the natural framework in which we formulate harmonic analysis on manifolds, heat flows and Onsager’s conjecture. Wherever there is the Laplacian, there is harmonic analysis. Historically, Milgram managed to establish a subset of Hodge theory by heat flow methods [MR51]. Here, however, we will establish Hodge theory by standard elliptic estimates, from which we develop analysis on manifolds and construct the heat flow. Most notably, Hodge theory greatly simplifies some crucial approximation steps involving the boundary (Corollary 70), and helps predict some key results Onsager’s conjecture would require (Theorem 15, Subsection 7.4, Subsection 8.3). That such leaps of faith turn out to be true only further underscore how well-made the conjecture is in its anticipation of undiscovered mathematics.

4 For those familiar with the smoothing properties of Littlewood-Paley projection as well as Bernstein t∆ inequalities [Tao06, Appendix A], the rough picture is that e ≈ P≤ √1 . While the introduction of t curvature necessitates the change of constants in estimates, and the boundary requires its own considerations, it is remarkable how far we can go with this analogy. Regarding the properties we will need for Onsager’s conjecture, there is a satisfying explanation: the theory of sectorial operators in functional analysis. This, together with Hodge theory, the theory of Besov spaces and interpolation theory, allows us to build a basic foundation for global analysis on Riemannian manifolds in general, which will be more than enough to handle Onsager’s conjecture. Hodge theory and sectorial operators, in their various forms, have been used in fluid dynamics for a long time by Fujita, Kato, Giga, Miyakawa et al. (cf. [FK64; Miy80; Gig81; GM85; BAE16] and their references). Although we will not use them for this paper, we also ought to mention the results regarding bisectorial operators, H∞ functional calculus, and Hodge theory on rough domains developed by Alan McIntosh, Marius Mitrea, Sylvie Monniaux et al. (cf. [McI86; DM96; FMM98; AM04; MM08; MM09a; MM09b; GMM10; She12; MM18] and their references), which generalize many Hodge-theoretic results in this paper. Alternative formalizations of Littlewood-Paley theory also exist (cf. [HMY08; KP14; FFP16; KW16; BBD18; Tan18] and their references). Here, we are mainly focused on the analogy between the heat flow and the Littlewood- Paley projection on Lp spaces of differential forms (over manifolds with boundary), as well as the interplay with Hodge theory. Lastly, we also introduce heatable currents – the largest space on which the heat flow can be profitably defined – as the analogue to tempered distributions on manifolds (Subsection 7.4). In doing so, we will realize that the energy-conserving weak solution in Onsager’s conjecture solves the Euler equation in the sense of heatable currents. This is an elegant insight that helps show how interconnected these subjects are. Much like how learning the language of measure theory can shed light on problems in calculus and familiarity with differential geometry simplifies many calculations in fluid dynamics, the cost of learning ostensibly complicated formalism is often dwarfed by the benefits in clarity it brings. That being said, accessibility is also important, and besides providing a gentle introduction to the theory with copious references, this paper also hopes to convince the reader of the naturality behind the formalism.

1.4 Blackboxes Since we draw upon many areas, the paper is intended to be as self-contained as possible, but we will assume familiarity with basic elements of functional analysis, harmonic analysis and complex analysis. Some familiarity with differential and Riemannian geometry is certainly needed (cf. [Lee09; Cha06]), as well as Penrose notation (cf. [Wal84, Section 2.4]). In addition, a number of blackbox theorems will be borrowed from the following sources:

1. For interpolation theory: Interpolation Spaces [BL76] and “Abstract Stein Interpolation” [Voi92]

2. For harmonic analysis and elements of functional analysis:

• Singular Integrals and Differentiability Properties of Functions. (PMS-30) [Ste71] • Partial Differential Equations I [Tay11a] • Recent developments in the Navier-Stokes problem (Chapman & Hall/CRC Research Notes in Mathematics Series) [Lem02]

3. For Besov spaces: Theory of Function Spaces; Theory of Function Spaces II [Tri10; Tri92]

4. For Hodge theory: Hodge Decomposition—A Method for Solving Boundary Value Problems [Sch95]

5 5. For semigroups and sectorial operators: One-parameter semigroups for linear evolution equations [Eng00] and Vector-valued Laplace Transforms and Cauchy Problems: Second Edition (Monographs in Mathematics) [Are+11]

The first three categories should be familiar with harmonic analysts.

1.5 For the specialists Some noteworthy characteristics of our approach:

• An alternative development of the (absolute Neumann) heat flow. In particular, the extrapolation of analyticity to Lp spaces does not involve establishing the resolvent estimate in Yosida’s half-plane criterion (Theorem 39), either via “Agmon’s trick” [Agm62] as done in [Miy80] or manual estimates as in [BAE16]. Instead, by abstract Stein interpolation, we only need the local boundedness of the heat flow on Lp, which can follow cleanly from Gronwall and integration by parts (Theorem 71). In short, functional analysis does the heavy lifting. We also managed to attain W 1,p-analyticity assuming the 1 p Neumann condition (Subsection 7.3), and Bp,1-analyticity via the Leray projection (Subsection 8.3). • We do not focus on the Stokes operator in this paper, but our results (Subsection 7.3, Subsection 8.3) do contain the case of the Stokes operator corresponding to the “Navier-type” / “free” boundary condition, as discussed in [Miy80; Gig82; MM09a; MM09b; BAE16] and others. This should not be confused with the Stokes operator corresponding to the “no-slip” boundary condition, as discussed in [FK64; GM85; MM08] and others. See [HS18] for more references.

• For simplicity, we stay within the smooth and compact setting, which, as Hilbert would say, is that special case containing all the germs of generality. An effort has also been made to keep the material concrete (as opposed to, for instance, using Hilbert complexes).

• Heatable currents are introduced as the analogue to tempered distributions, and we show how they naturally appear in the characterization of the adjoints of d and δ (Subsection 7.4).

• A refinement of a special case of the fractional Leibniz rule, with the supports of functions taken into account, is given in Theorem 54.

• For the proof of Onsager’s conjecture, there are some subtle, but substantial differences with [IO15]:

– In [IO15], Besov spaces are defined by the heat flow, and compatibility with the usual scalar Besov spaces is proven when M is Rn or Tn. Here we will use the standard scalar Besov spaces as defined by Triebel in [Tri10; Tri92], and prove the appropriate estimates for the heat flow by interpolation. – The heat flow used by Isett & Oh (constructed by Strichartz using dissipative operators) is gen- erated by the Hodge Laplacian, which is self-adjoint in the no-boundary case. In the case with boundary, there are four different self-adjoint versions for the Hodge Laplacian (see Theorem 61), and we choose the absolute Neumann version. There are also heat flows generated by the connection Laplacian, but we do not use them in this paper since the connection Laplacian does not commute with the exterior derivative and the Leray projection etc. The theory of dissipative operators is also not sufficient to establish Lp-analyticity and W 1,p-analyticity for all p ∈ (1, ∞), so we instead use the theory of sectorial operators, which is made for this purpose.

6 – The commutator we will use is a bit different from that in [IO15]. This will help us eliminate some boundary terms. We will also avoid the explicit formula and computations in [IO15, Lemma 4.4], as they also lead to various boundary terms. Generally speaking, the covariant derivative behaves badly on the boundary.

• A calculation of the pressure by negative-order Hodge-Sobolev spaces (Subsection 8.2).

• More results will be proven for analysis on manifolds than needed for Onsager’s conjecture, as they are of independent interest. For the sake of accessibility, we will also review most of the relevant background material, with the assumption that the reader is a harmonic analyst who knows some differential geometry. It is hard to overstate our indebtedness to all the mathematicians whose work our theory will build upon, from harmonic analysis to Hodge theory and sectorial operators, and yet hopefully each will be able to find within this paper something new and interesting.

2 Common notation

It might not be an exaggeration to say the main difficulty in reading a paper dealing with Hodge theory is understanding the notation, and an effort has been made to keep our notation as standard and self- explanatory as possible. Some common notation we use:

• A .x,¬y B means A ≤ CB where C > 0 depends on x and not y. Similarly, A ∼x,¬y B means A .x,¬y B and B .x,¬y A. When the dependencies are obvious by context, we do not need to make them explicit.

• N0, N1 : the set of natural numbers, starting with 0 and 1 respectively. • DCT: dominated convergence theorem, FTC: fundamental theorem of calculus, PTAS: passing to a subsequence, WLOG: without loss of generality.

• TVS: topological vector space, NVS: normed vector space, SOT: strong operator topology.

• For TVS X, Y ≤ X means Y is a subspace of X.

•L (X,Y ) : the space of continuous linear maps from TVS X to Y . Also L(X) = L(X,X).

• C0(S → Y ): the space of bounded, continuous functions from metric space S to normed vector space 0 Y . Not to be confused with Cloc(S → Y ), which is the space of locally bounded, continuous functions. ∗ •k xkD(A) = kxkX +kAxkX and kxkD(A) = kAxkX where A is an unbounded operator on (real/complex) ∗ ∞ Banach space X and x ∈ D(A). Note that k·kD(A) is not always a norm. Also define D(A ) = k ∩k∈N1 D(A ). + − + • For δ ∈ (0, π], define the open sector Σδ = {z ∈ C\{0} : | arg z| < δ},Σδ = −Σδ , D = {z ∈ C : |z| < + − + 1}. Also define Σ0 = (0, ∞) and Σ0 = −Σ0 . • B(x, r): the open ball of radius r centered at x in a metric space.

•S (Rn): the space of Schwartz functions on Rn, S(Ω): restrictions of Schwartz functions to the domain Ω ⊂ Rn. There is also a list of other symbols we will use at the end of the paper.

7 3 Onsager’s conjecture

3.1 Summary of preliminaries At the cost of some slight duplication of exposition, we will quickly summarize the key tools we need for the proof, and leave the development of such tools for the rest of the paper. Alternatively, the reader can read the theory first and come back to this section later.

Definition 1. For the rest of the paper, unless otherwise stated, let M be a compact, smooth, Riemannian n-dimensional manifold, with no or smooth boundary. We also let I ⊂ be an open time interval. We write R ◦

M 0 small. Similarly define M≥r,Misomorphism, we can consider XM (the space of smooth vector fields) mostly the same 1 as Ω (M) (the space of smooth 1-forms), mutatis mutandis. We note that XM, X (∂M) and XM ∂M are different. Unless otherwise stated, let the implicit domain be M, so X stands for XM, and similarly Ωk for ΩkM. For X ∈ X, we write X[ as its dual 1-form. For ω ∈ Ω1, we write ω] as its dual vector field. ◦ k Let X00 (M) denote the set of smooth vector fields of compact support in M. Define Ω (M) similarly ◦ 00 (smooth differential forms with compact support in M). Let ν denote the outwards unit normal vector field on ∂M. ν can be extended via geodesics to a smooth vector field νe which is of unit length near the boundary (and cut off at some point away from the boundary). For X ∈ XM, define nX = hX, νi ν ∈ XM| (the normal part) and tX = X| − nX (the ∂M ∂M tangential part). We note that tX and nX only depend on X ∂M , so t and n can be defined on XM ∂M , ∼ and t (XM|∂M ) −→ X(∂M). For ω ∈ Ωk (M) , define tω and nω by

tω(X1, ..., Xk) := ω(tX1, ..., tXk) ∀Xj ∈ XM, j = 1, ..., k

[ [ and nω = ω|∂M − tω. Note that (nX) = nX ∀X ∈ X. Let ∇ denote the Levi-Civita connection, d the exterior derivative, δ the codifferential, and ∆ = − (dδ + δd) the Hodge-Laplacian, which is defined on vector fields by the musical isomorphism. p m,p s Familiar scalar function spaces such as L ,W (Lebesgue-Sobolev spaces), Bp,q (Besov spaces), C0,α (Holder spaces) (see Section5 for precise definitions) can be defined on M by partitions of unity and given a unique topology (Subsection 5.2, Subsection 6.1.2). Similarly, we define such function spaces for tensor fields and differential forms on M by partitions of unity and local coordinates (see subsection 6.1). 1 2 3 For instance, we can define L X or B3,1X.

1 1 1
1,p p p 0,α α α 3 Fact 2. ∀α ∈ 3 , 1 , ∀p ∈ (1, ∞): W X ,→ Bp,1X ,→L X and C X = B∞,∞X ,→ B3,∞X ,→ B3,1X (cf. Subsection 5.2, Subsection 5.4)

Definition 3. We write h·, ·i to denote the Riemannian fiber metric for tensor fields on M. We also define the dot product Z hhσ, θii = hσ, θi vol M where σ and θ are tensor fields of the same type, while vol is the Riemannian volume form. When there is no possible confusion, we will omit writing vol. k We define XN = {X ∈ X : nX = 0 } (Neumann condition). Similarly, we can define ΩN . In order to define the Neumann condition for less regular vector fields (and differential forms), we need to use the trace theorem.

8 Fact 4. (Subsection 5.2, Subsection 6.1.2) Let p ∈ [1, ∞). Then

1 1 p p p p • Bp,1 (M)  L (∂M) and Bp,1XM  L XM ∂M are continuous surjections.

m+ 1 p m m,p •∀ m ∈ N1 : Bp,1 XM  Bp,1XM ∂M ,→ W XM ∂M is continuous. Also closely related is the coarea formula:

1 p Fact 5. (Theorem 53) Let p ∈ [1, ∞), r > 0 be small and f be in Bp,1(M):   1. [0, r) → , ρ 7→ kfk p is continuous and bounded by C kfk 1 for some C > 0. R L (∂M>ρ) p Bp,1

2. |Mρ) p Lρ((0,r))

r↓0 3. kfk p kfk 1 and kfk p −−→kfk p , where avg means normalizing L (M≤r ,avg) .¬r p L (M≤r ,avg) L (∂M,avg) Bp,1(M) the measure to make it a probability measure.

1 p p r↓0 4. Let f ∈ L (I → B (M)), then kfk 1 ¬r kfk p p −−→kfk p p . p,1 p p & Lt L (M≤r ,avg) Lt L (∂M,avg) Lt Bp,1(M)

1 p Analogous results hold if f ∈ Bp,1X. (Subsection 6.1.2)

1 1 3 3 1,3 Therefore, we can define spaces such as B3,1XN = {X ∈ B3,1X : nX = 0 } and W XN . However, 2 2 something like L XN would not make sense since the trace map does not continuously extend to L X.

Definition 6. We define P as the Leray projection (constructed in Theorem 68), which projects X onto   Ker div . Note that the Neumann condition is enforced by . XN P

m,p m,p m,p    Fact 7. ∀m ∈ 0, ∀p ∈ (1, ∞), is continuous on W X and (W X) = W -cl Ker div N P P XN (closure in the W m,p-topology). (Subsection 6.4)

We collect some results regarding our heat flow in one place:

p Fact 8 (Absolute Neumann heat flow). There exists a semigroup of operators (S(t))t≥0 acting on ∪p∈(1,∞)L X such that

1. S (t1) S (t2) = S (t1 + t2) ∀t1, t2 ≥ 0 and S (0) = 1.

2. (Subsection 7.2) ∀p ∈ (1, ∞) , ∀X ∈ LpX :

(a) S(t)X ∈ XN and ∂t (S(t)X) = ∆S(t)X ∀t > 0. C∞ (b) S(t)X −−−→ S (t0) X ∀t0 > 0. t→t0 m 1
2 (c) kS(t)XkW m,p .m,p t kXkLp ∀m ∈ N0, ∀t ∈ (0, 1). p (d) S(t)X −−−→L X. t→0

1,p 3. (Subsection 7.3) ∀p ∈ (1, ∞) , ∀X ∈ W XN :

m 1
2 (a) kS(t)XkW m+1,p .m,p t kXkW 1,p ∀m ∈ N0, ∀t ∈ (0, 1).

9 1,p (b) S(t)X −−−→W X. t→0 m,p 4. (Theorem 76) S (t) P = PS (t) on W X ∀m ∈ N0, ∀p ∈ (1, ∞) , ∀t ≥ 0.

0 5. (Subsection 7.2) hhS(t)X,Y ii = hhX,S(t)Y ii ∀t ≥ 0, ∀p ∈ (1, ∞) , ∀X ∈ LpX, ∀Y ∈ Lp X.

t∆ These estimates precisely fit the analogy e ≈ P≤ √1 where P is the Littlewood-Paley projection. t We also stress that the heat flow preserves the space of tangential, divergence-free vector fields (the range of P), and is intrinsic (with no dependence on choices of local coordinates). Analogous results hold for scalar functions and differential forms (Section7). We also have commutativity with the exterior derivative and codifferential in the case of differential forms (Theorem 73). Loosely speaking, this allows the heat flow to preserve the overall Hodge structure on the manifold. All these properties would not be possible under standard mollification via partitions of unity. Note that for X ∈ X, X ⊗ X is not dual to a differential form. As our heat flow is generated by the Hodge Laplacian, it is less useful in mollifying general tensor fields (for which the connection Laplacian is better suited). Fortunately, we will never actually have to do so in this paper. We observe some basic identities (cf. Theorem 58):

• Using Penrose abstract index notation (see Subsection 6.2), for any smooth tensors Ta1...ak , we define (∇T ) = ∇ T and div T = ∇iT . ia1...ak i a1...ak ia2...ak

• For all smooth tensors Ta1...ak and Qa1...ak+1 : Z Z Z Z ia1...ak
ia1...ak ia1...ak ia1...ak ∇i Ta1...ak Q = ∇iTa1...ak Q + Ta1...ak ∇iQ = νiTa1...ak Q M M M ∂M

∞ • For X ∈ XN ,Y ∈ X, f ∈ C (M): R R R R R R 1. M Xf = M div (fX) − M fdiv (X) = ∂M hfX, νi − M fdivX = − M fdivX R R 2. M hdiv(X ⊗ X),Y i = − M hX ⊗ X, ∇Y i

ij i σj j iσ σ ij σ ij ij • (∇a∇b − ∇b∇a) T kl = −Rabσ T kl − Rabσ T kl + Rabk T σl + Rabl T kσ for any tensor T kl, where R is the Riemann curvature tensor. Similar identities hold for other types of tensors. When we do not care about the exact indices and how they contract, we can just write the schematic ij identity (∇a∇b − ∇b∇a) T kl = R ∗ T. As R is bounded on compact M, interchanging derivatives is a zeroth-order operation on M. In particular, we have the Weitzenbock formula:

i ∆X = ∇i∇ X + R ∗ X ∀X ∈ XM (2)

• For X ∈ PL2X,Y ∈ X,Z ∈ X, f ∈ C∞ (M): R 1. M Xf = 0 R R 2. M h∇X Y,Zi = − M hY, ∇X Zi . There is an elementary lemma which is useful for convergence (the proof is straightforward and omitted):

Lemma 9 (Dense convergence). Let X,Y be (real/complex) Banach spaces and X0 ≤ X be norm-dense.

Let (Tj)j∈N be bounded in L(X,Y ) and T ∈ L(X,Y ). If Tjx0 → T x0 ∀x0 ∈ X0 then Tjx → T x ∀x ∈ X.

10 ◦ Definition 10 (Heatable currents). As the heat flow does not preserve compact supports in M, it is not defined on distributions. This inspires the formulation of heatable currents. Define:

◦ k k k ∞
• DΩ = Ω00 = colim{ Ω00 (K) ,C topo : K ⊂ M compact} as the space of test k-forms with Schwartz’s topology1 (colimit in the category of locally convex TVS).

0 k k
∗ • D Ω = DΩ as the space of k-currents (or distributional k-forms), equipped with the weak* topology.

k k m m • DN Ω = {ω ∈ Ω : n∆ ω = 0, nd∆ ω = 0 ∀m ∈ N0} as the space of heated k-forms with the ∞ 0 k k
∗ Frechet C topology and DN Ω = DN Ω as the space of heatable k-currents (or heatable distributional k-forms) with the weak* topology.

k
∞ k
∞ k
∞
• Spacetime test forms: D I, Ω = Cc I, Ω00 = colim{ Cc I1, Ω00(K) ,C topo : I1 × K ⊂ ◦ k
∞ k
∞
I × M compact} and DN I, Ω = colim{ Cc I1, DN Ω ,C topo : I1 ⊂ I compact}.

0 k
k
∗ 0 k
k
∗ • Spacetime distributions D I, Ω = D I, Ω , DN I, Ω = DN I, Ω . 1 In particular, DN X is defined from DN Ω by the musical isomorphism, and it is invariant under our heat flow (much like how the space of Schwartz functions S(Rn) is invariant under the Littlewood-Paley projection). By that analogy, heatable currents are tempered distributions on manifolds, and we can write

0 hhS(t)Λ,Xii = hhΛ,S (t) Xii ∀Λ ∈ DN X, ∀X ∈ DN X, ∀t ≥ 0

where the dot product hh·, ·ii is simply abuse of notation.

0 Fact 11. Some basic properties of DN X and DN X:

• hh∆X,Y ii = hhX, ∆Y ii ∀X,Y ∈ DN X. (Theorem 58)

0 • S(t)Λ ∈ DN X ∀t > 0, ∀Λ ∈ DN X. (Subsection 7.4, a heatable current becomes heated once the heat flow is applied)

p p 0 • X00 ⊂ DN X and is dense in L X ∀p ∈ [1, ∞). Also, L X ,→ DN X is continuous ∀p ∈ [1, ∞].

1 1 3 3 1,p 1,p • PB3,1X = PB3,1XN , PW X = PW XN and PDN X ≤ DN X. (Subsection 6.4)

1 1 1,p 1,p 3 3 • W -cl (DN X) = W XN ∀p ∈ (1, ∞) (Subsection 7.3), B3,1-cl (PDN X) = PB3,1XN (Subsection 8.3)

C∞ •∀ X ∈ DN X : S(t)X −−→ X and ∂t (S(t)X) = ∆S(t)X = S(t)∆X ∀t ≥ 0. (Theorem 32, Subsec- t↓0 tion 7.2)

0 • (Subsection 7.2, Subsection 8.3) ∀t ∈ (0, 1), ∀m, m ∈ N0, ∀p ∈ (1, ∞), ∀X ∈ DN X :

m0 1
2 1. kS(t)XkW m+m0,p . t kXkW m,p . 1 m0 1
2p + 2 2. kS(t)Xk m+m0+ 1 kXk m,p . p . t W Bp,1 1 1 2 (m− p ) 3. t kS(t)Xk m,p + kS(t)Xk 1 kXk 1 when m ≥ 1 and X ∈ X. W p . p PDN Bp,1 Bp,1 1 B 3 1 3,1 3 By dense convergence (Lemma9), this means S(t)X −−−→ X ∀X ∈ PB3,1XN . t↓0 1Confusingly enough, “Schwartz’s topology” refers to the topology on the space of distributions, not the topology for Schwartz functions.

11 1 1 s↓0 3 3 Corollary 12 (Vanishing). ∀X ∈ PB3,1XN : s kS(s)XkW 1,3 −−→ 0.

1 1 3 3 s↓0 Remark. So, for U ∈ L B XN : kU(t)k 1 σ 3 kS(σ)U(t)k 1,3 −−−→ 0. t P 3,1 3 3 & W ∞ Lt B3,1 Lσ ([0,s]) 3 DCT Lt This pointwise vanishing property becomes important for the commutator estimate in Onsager’s conjecture 1 at the critical regularity level 3 , while higher regularity levels have enough room for vanishing in norm (which is better).

1 1 s↓0 3 3 Proof. For Y ∈ PDN X, as s > 0 small: s kS(s)Y kW 1,3 . s kY kW 1,3 −−→ 0. Then note 1 1 3 s 3 kS(s)Xk 1,3 kXk 1 ∀X ∈ B X , so we can apply dense convergence (Lemma9). W . 3 P 3,1 N B3,1

3.2 Searching for the proper formulation Onsager’s conjecture states that energy is conserved when V has enough regularity, with appropriate condi- tions near the boundary. But making this statement precise is half of the challenge. Definition 13. We say (V, p) is a weak solution to the Euler equation when

2 2
1 •V∈ Lloc I, PL X , p ∈ Lloc(I × M) ∞ RR • ∀X ∈ Cc (I, X00): I×M hV, ∂tX i + hV ⊗ V, ∇X i + p div X = 0.

The last condition means ∂tV + div(V ⊗ V) + grad p = 0 as spacetime distributions. Note that V ⊗ V ∈ 1 1
Lloc I,L X so it is a distribution. The keen reader should notice we use a different font for time-dependent vector fields. There is not enough time-regularity for FTC, and we cannot say

Z t1 Z t1 Z hhV (t1) ,Xii − hhV (t0) ,Xii = hhV ⊗ V, ∇Xii + p div X ∀X ∈ X00 t0 t0 M

But we can still use approximation to the identity (in the time variable) near t0,t1, as well as Lebesgue differentiation to get something similar for a.e. t0, t1. By using dense convergence (Lemma9) and modifying 0 2

∞ 2
I into I0 ⊂ I such that |I\I0| = 0, we can say V ∈ Cloc I0, L X, weak ≤ Lloc I,L X . 0 2
 2  We do not have V ∈ Cloc I,L X , so energy conservation only means ∂t kV(t)kL2X = 0 as a distribu- tion. In other words, the goal is to show Z 0 ∞ η (t) hhV(t), V(t)ii dt = 0 ∀η ∈ Cc (I) I

∞ Next, having the test vector field X ∈ C (I, X00) can be quite restrictive, since the heat flow (much like c ◦ the Littlewood-Paley projection) does not preserve compact supports in M. We need a notion that is more in tune with our theory.

2 2
Definition 14. We say (V, p) is a Hodge weak solution to the Euler equation when V ∈ Lloc I, PL X , 1 p ∈ Lloc(I × M) and ZZ ∞ ∀X ∈ Cc (I, XN ): hV, ∂tX i + hV ⊗ V, ∇X i + p div X = 0 I×M

Now this looks better, since XN is invariant under the heat flow. However, this is a leap of faith we will need to justify later (cf. Subsection 3.3).

12 As PX ≤ XN , we can go further and say V is a Hodge-Leray weak solution to the Euler equation 2 2
when V ∈ Lloc I, PL X and ZZ ∞ ∀X ∈ Cc (I, PX): hV, ∂tX i + hV ⊗ V, ∇X i = 0 I×M

This would help give a formulation of Onsager’s conjecture that does not depend on the pressure, similar to [RRS18].

3 0,α Next, we look at the conditions for V and p near ∂M. In [BT17], they assumed V ∈ Lt C XN with 1
3 α 1
α ∈ 3 , 1 . In [NN18], they assumed V ∈ Lt B3,∞X (α ∈ 3 , 1 ) with a more general “strip decay” condition:

2 r↓0 • kVk 3 3 khV, νik 3 3 −−→ 0 Lt L (M

r↓0 •k pk 3 3 khV, νikL3L3(M ,avg) −−→ 0. 2 2 e t

3 1/3 In [Bar+19] (the most recent result), they assumed V ∈ Lt B3,VMOX (see the paper for the full definition), along with a minor relaxation for the “strip decay” condition:

2 ! |V| r↓0 + p hV, νi −−→ 0 2 e   1 1 L L M[ r , r ],avg t 4 2

1 3 3 r↓0 When V ∈ L B X, khV, νik 3 3 −−→ khV, νik 3 3 by Fact5. This motivates our t 3,1 e Lt L (M

3.3 Justification of formulation We define the cutoffs

ψr(x) = Ψr (dist (x, ∂M)) (3)

∞ 0 1 where r > 0 small, Ψr ∈ C ([0, ∞), [0, ∞)) such that 1 3 ≥ Ψr ≥ 1[0, r ] and kΨ k . [0, 4 r) 2 r ∞ . r 1 Then ∇ψr(x) = fr(x)νe(x) where |fr(x)| . r and supp ψr ⊂ M

3 α 2 r↓0 2. V ∈ L B X and kVk 3 3 khV, νik 3 3 −−→ 0. t 3,∞ Lt L (M

1 3 3 3. V ∈ Lt B3,1XN . 4.( V, p) is a Hodge weak solution.

5. V is a Hodge-Leray weak solution.

Theorem 15. We have (1) =⇒ (2) =⇒ (3) =⇒ (4) =⇒ (5).

1 0,α α α 3 Proof. By Fact2, C XN = B∞,∞XN ,→ B3,∞XN ,→ B3,1XN . Then by the coarea formula,

3 2 2 khV, νik 3 3 kVk 3 3 khV, νik 3 3 kVk 1 khV, νik 3 3 e Lt L (M

13 1 3 3 2 r↓0 So for V ∈ L B X: kVk 3 3 khV, νik 3 3 −−→ 0 ⇐⇒ khV, νik 3 3 = t 3,1 Lt L (M

As (4) =⇒ (5) is obvious, the only thing left is to show (3) =⇒ (4). Recall the cutoffs ψr from Equation (3). ∞ ∞ Let I1 ⊂ I be bounded and X ∈ Cc (I1, XN ), then (1 − ψr) X ∈ Cc (I, X00), and so by the definition of weak solution: ZZ 0 = (1 − ψr) hV, ∂tX i + hV, ∇V ((1 − ψr) X )i + p div ((1 − ψr) X ) I×M ZZ ZZ = (1 − ψr)(hV, ∂tX i + hV, ∇V X i + p div X ) − (hV, ∇ψri hV, X i + p hX , ∇ψri) I×M I×M

We are done if the first term goes to zero as r ↓ 0 . So we only need to show the second term goes to zero. Since ∇ψr = frνe and supp ψr ⊂ M

fr hV, νei hV, X i + pfr hX , νei I1×M

kVk 3 3 khV, νik 3 3 kX k 3 3 + kpk 1 khX , νik ∞ 0,1 . Lt L (M

We used the estimate khX , νik ∞ r khX , νik 0,1 since hX , νi = 0 on ∂M. e L (M

Remark. Interestingly, as Subsection 3.5 will show, no “strip decay” condition involving p seems to be necessary. See the end of Subsection 1.1 for a discussion of this minor improvement. ◦ We briefly note that when ∂M = ∅, it is customary to set dist (x, ∂M) = ∞, and ψr = 0, M>r = M = M, M

3.4 Heating the nonlinear term

1 3 1 Let U, V ∈ B3,1X. Then U ⊗ V ∈ L X and div (U ⊗ V ) is defined as a distribution. To apply the heat flow to div (U ⊗ V ), we need to define (div (U ⊗ V ))[ so that it is heatable. Recall integration by parts: Z hhdiv (Y ⊗ Z) ,Xii = − hhY ⊗ Z, ∇Xii + hν, Y i hZ,Xi ∀X,Y,Z ∈ X (M) ∂M R Observe that for X ∈ X, even though hhdiv (U ⊗ V ) ,Xii is not defined, ∂M hν, Ui hV,Xi − hhU ⊗ V, ∇Xii is well-defined by the trace theorem. So we will define the heatable 1-current (div (U ⊗ V ))[ by Z hhdiv (U ⊗ V ) ,Xii = − hhU ⊗ V, ∇Xii + hν, Ui hV,Xi ∀X ∈ DN X (X is heated) ∂M

It is continuous on X since |hhdiv (U ⊗ V ) ,Xii| kUk 1 kV k 1 kXk 1 + kUk 3 kV k 3 k∇Xk 3 . By DN . 3 3 3 L L L B3,1 B3,1 B3,1 the same formula and reasoning, we see that (div (U ⊗ V ))[ is not just heatable, but also a continuous linear functional on (X (M) ,C∞ topo).

14 On the other hand, we can get away with less regularity by assuming U ∈ PL2X. Then we simply need to define hhdiv (U ⊗ V ) ,Xii = − hhU ⊗ V, ∇Xii ∀X ∈ X. [ In short, (div (U ⊗ V )) is heatable when U ∈ PL2X and V ∈ L2X. Consequently, by Theorem 15, when 1 3 3 (V, p) is a weak solution to the Euler equation and V ∈ Lt B3,1XN :(V, p) is a Hodge weak solution and

0 ∂tV + div(V ⊗ V) + grad p = 0 in DN (I, X) . (4)

3.5 Proof of Onsager’s conjecture For the rest of the proof, we will write et∆ for S(t), as we will not need another heat flow. For ε > 0 and vector field X, we will write Xε for eε∆X. We opt to formulate the conjecture without mentioning the pressure (see Subsection 3.3 for the justifi- cation).

Theorem 16 (Onsager’s conjecture). Let M be a compact, oriented Riemannian manifold with no or smooth 1 3 3 ∞ RR boundary. Let V ∈ Lt PB3,1XN such that ∀X ∈ Cc (I, PX): I×M hV, ∂tX i+hV ⊗ V, ∇X i = 0 (Hodge-Leray weak solution). Then we can show Z 0 ∞ η (t) hhV(t), V(t)ii dt = 0 ∀η ∈ Cc (I) I Consequently, hhV(t), V(t)ii is constant for a.e. t ∈ I.

As usual, there is a commutator estimate which we will leave for later:

Z DD EE Z η div (U ⊗ U)2ε , U 2ε − η div U 2ε ⊗ U 2ε
, U 2ε I I Z DD EE Z DD ε EE ε↓0 = η div (U ⊗ U)3ε , U ε − η div U 2ε ⊗ U 2ε
, U ε −−→ 0 (5) I I

1 3 3 ∞ for all U ∈ Lt PB3,1XN , η ∈ Cc (I). Notation: we write div (U ⊗ U)ε for (div (U ⊗ U))ε and ∇U ε for ∇ (U ε) (recall that the heat flow does not work on tensors U ⊗ U and ∇U). Compared with [IO15], our commutator estimate looks a bit different, to ease some integration by parts procedures down the line. [ Remark. For any U in PL2X, div (U ⊗ U) is a heatable 1-current (see Subsection 3.4). In particular, for ε > 0, div (U ⊗ U)ε is smooth and

hhdiv (U ⊗ U)ε ,Y ii = − hhU ⊗ U, ∇ (Y ε)ii ∀Y ∈ X (6)

Consequently, Equation (5) is well-defined.

R 0 Theorem 17 (Onsager). Assume Equation (5) is true. Then I η (t) hhV(t), V(t)ii dt = 0.

∞ τ↓0 ε ε∆ Proof. Let Φ ∈ Cc (R) and Φτ −−→ δ0 be a radially symmetric mollifier. Write V for e V (spatial mollification) and Vτ for Φτ ∗ V (temporal mollification). First, we mollify in time and space Z Z 1 0 DCT 1 0 ε ε η hhV, Vii = lim lim η hhVτ , Vτ ii 2 I ε↓0 τ↓0 2 I

15 Then we want to get rid of the time derivative: Z Z Z Z 1 0 ε ε ε ε ε ε 0 ε ε η hhVτ , Vτ ii = − η hh∂tVτ , Vτ ii = − hh∂t (ηVτ ) , Vτ ii + η hhVτ , Vτ ii 2 I I I I Then we use the definition of Hodge-Leray weak solution, and exploit the commutativity between spatial and temporal operators: Z Z Z 1 0 ε ε ε ε  2ε
 η hhVτ , Vτ ii = hh∂t (ηVτ ) , Vτ ii = ∂t ηVτ τ , V 2 I I I Z  2ε
 2ε
∞ = − ∇ ηVτ τ , V ⊗ V as ηVτ τ ∈ Cc (I, PX) I Z  2ε
 = − η ∇Vτ τ , V ⊗ V I Z 2ε
= − η ∇V τ , (V ⊗ V)τ I

As there is no longer a time derivative on V, we get rid of τ by letting τ ↓ 0 (fine as V is L3 in time). Recall Equation (6):

1 Z Z Z η0 hhVε, Vεii = − η ∇ V2ε
, V ⊗ V = η hhVε, div (V ⊗ V)εii 2 I I I Z ε ε ε = η hhV , div (V ⊗ V )ii + oε(1) (commutator estimate) I Z Z Z ε 2 ! ε ε ε |V | ε = η hhV , ∇Vε V ii + oε(1) = η V + oε(1) = oε(1) as V ∈ PX I I M 2

1 R 0 1 R 0 ε ε 1 R 0 ε ε So 2 I η hhV, Vii = limε↓0 limτ↓0 2 I η hhVτ , Vτ ii = limε↓0 2 I η hhV , V ii = 0.

The proof is short and did not much use the Besov regularity of V. It is the commutator estimate that presents the main difficulty. We proceed similarly as in [IO15]. 1 1 3 3 3 Let U ∈ Lt PB3,1XN . By setting U(t) to 0 for t in a null set, WLOG U(t) ∈ PB3,1XN ∀t ∈ I. Define the commutator  s W(t, s) = div (U(t) ⊗ U(t))3s − div U (t)2s ⊗ U (t)2s

3s When t and s are implicitly understood, we will not write them. As div (U(t) ⊗ U(t)) solves (∂s − 3∆) X = 0, we define N = (∂s − 3∆) W. Then W and N obey the Duhamel formula:

Lemma 18 (Duhamel formulas).

s↓0 0 0 s↓0 0 1. W(t, s) −−→ 0 in DN X and therefore in D X. Furthermore, W(·, s) −−→ 0 in DN (I, X) and therefore in D 0 (I, X) (spacetime distribution).

R s 3(s−σ) ε↓0 0 2. For fixed t0 ∈ I and s > 0: ε N (t0, σ) dσ −−→W (t0, s) in DN X.

Proof.

16 ∞ 1. Let X ∈ DN X, X ∈ Cc (I, DN X) . It is trivial to check (with DCT)

s↓0 U(t) ⊗ U(t), ∇ X3s
− U(t)2s ⊗ U(t)2s, ∇ (Xs) −−→ 0

Z Z s↓0 U ⊗ U, ∇ X 3s
− U 2s ⊗ U 2s, ∇ (X s) −−→ 0 I I

s∆ 0 2. Let ε > 0. By the smoothing effect of e , W(t0, ·) and N (t0, ·) are in Cloc ((0, 1], DN X). As s∆
m e s≥0 is a C0 semigroup on (H -cl (DN X) , k·kHm ) ∀m ∈ N0, and a semigroup basically corresponds to an ODE (cf. [Tay11a, Appendix A, Proposition 9.10 & 9.11]), from ∂sW = 3∆W + N for s ≥ ε we get the Duhamel formula

Z s 3(s−ε) 3(s−σ) ∀s > ε : W(t0, s) = W (t0, ε) + N (t0, σ) dσ ε

0 3(s−ε) DN X So we only need to show W (t0, ε) −−−→0. Let X ∈ DN X. ε↓0

ε DD 3(s−ε)EE DD 3(s−ε) 3εEE DD 3(s−ε)  2ε 2ε EE X, W (t0, ε) = X , div (U (t0) ⊗ U (t0)) − X , div U (t0) ⊗ U (t0)

3s
DD 3s−2ε
2ε 2εEE ε↓0 = − ∇ X , U (t0) ⊗ U (t0) + ∇ X , U (t0) ⊗ U (t0) −−→ 0.

R s R s From now on, we write 0+ for limε↓0 ε . Then

Z Z Z s DD EE dt η (t) hhW (t, s) , U (t)sii = dt η (t) dσ N (t, σ)3(s−σ) , U (t)s I I 0+

To clean up the algebra, we will classify the terms that are going to appear but are actually negligible in the end. The following estimates lie at the heart of the problem, showing why the regularity needs to be at 1 least 3 , and that our argument barely holds thanks to the pointwise vanishing property (Corollary 12).

1 k ! 2 P (j) 2 Lemma 19 (3 error estimates). Define the k-jet fiber norm |X|J k = ∇ X ∀X ∈ X (more j=0 details in Subsection 6.1.1). Then we have

R R s R 2σ 2 4s−2σ s↓0 1. I |η| 0+ dσ M U J 1 U J 1 −−→ 0

R R s R 2σ 2 4s−2σ s↓0 2. I |η| 0+ dσ ∂M U U J 2 −−→ 0

R R s R 2σ 2σ 4s−2σ s↓0 3. I |η| 0+ dσ ∂M U U J 1 U J 1 −−→ 0

1 s s 1 s 3 2 1
6 2 Proof. Define A (t, s) = s U (t) . Then for s > 0 small: kU (t) k 1+ 1 U (t) 1,3 3 . s 1,3 . W B3,1 W 1 1
2 s↓0 s A (t, s) and kA (t, σ)k ∞ −−→ 0 by Corollary 12. We also note that kU (t) k 2+ 1 s Lσ≤s 3 3 . Lt B3,1 2 s 1
3 2 1
s U (t) . s A (t, s). W 1,3 Now we can prove the error estimates go to 0:

17 1. Z Z s Z Z Z s 2σ 2 4s−2σ 2σ 2 4s−2σ |η| dσ U J 1 U J 1 . |η| dσ U W 1,3 U W 1,3 I 0+ M I 0+ 2 1 Z Z s   3   3 1 1 2 . dt |η(t)| dσ A (t, 2σ) A (t, 4s − 2σ) I 0+ σ 2s − σ 2 1 Z Z 1   3   3 Z σ7→sσ 1 1 2 3 s↓0 = dt |η(t)| dσ A (t, 2sσ) A (t, 4s − 2sσ) . dt |η(t)| kA (t, σ)kL∞ −−→ 0. I 0+ σ 2 − σ I σ≤4s

2. Z Z s Z Z Z s 2σ 2 4s−2σ 2σ 2 4s−2σ |η| dσ U U J 2 . |η| dσ U L3XM| U W 2,3XM| I 0+ ∂M I 0+ ∂M ∂M s Trace Z Z 2 2σ 1 4s−2σ |η| dσ U U 2+ 1 . B 3 XM 3 I 0+ 3,1 B3,1 XM Z Z s   2 1 dt |η (t)| kU (t)k 1 dσ A (t, 4s − 2σ) . 3 I B3,1XM 0+ 2s − σ Z Z 1   Z σ7→sσ 2 1 2 = dt |η (t)| kU (t)k 1 dσ A (t, 4s − 2sσ) dt |η (t)| kU (t)k 1 kA (t, σ)k ∞ 3 . 3 Lσ≤4s I B3,1XM 0+ 2 − σ I B3,1XM 2 s↓0 kUk 1 kA (t, σ)k ∞ −−→ 0 . 3 3 Lσ≤4s 3 Lt B3,1(M) Lt

3. Z Z s Z 2σ 2σ 4s−2σ |η| dσ U U J 1 U J 1 I 0+ ∂M Trace Z Z s 2σ 2σ 4s−2σ |η| dσ U 1 U 1+ 1 U 1+ 1 . 3 3 3 I 0+ B3,1XM B3,1 XM B3,1 XM 1 1 Z Z s  1  2  1  2 dt |η (t)| kU (t)k 1 dσ A (t, 2σ) A (t, 4s − 2σ) . 3 I B3,1 0+ σ 2s − σ 1 1 1   2   2 σ7→sσ Z Z 1 1 = dt |η (t)| kU (t)k 1 dσ A (t, 2sσ) A (t, 4s − 2sσ) 3 I B3,1 0+ σ 2 − σ Z 2 2 s↓0 dt |η (t)| kU (t)k 1 kA (t, σ)k ∞ kUk 1 kA (t, σ)k ∞ −−→ 0 . 3 Lσ≤4s . 3 3 Lσ≤4s 3 I B3,1 Lt B3,1 Lt

Note that

 2σ 2σ
σ 2σ 2σ
σ 2σ 2σ
σ 2σ 2σ
σ N (t, σ) = (∂σ − 3∆) − div U ⊗ U = −2 div ∆U ⊗ U − 2 div U ⊗ ∆U + 2∆ div U ⊗ U

Finally, we will show

Z Z s↓0 η hhW(s), U sii = dt η (t) hhW(t, s), U (t)sii −−→ 0 I I

18 Proof. Integrate by parts into 3 components:

Z Z Z s DD EE Z Z s DD EE η hhW(s), U sii = dt η (t) dσ N (t, σ)3(s−σ) , U (t)s = dt η (t) dσ N (t, σ) , U (t)4s−3σ I I 0+ I 0+ Z Z s Z Z s = 2 η dσ ∆U 2σ ⊗ U 2σ, ∇ U 4s−2σ
+ 2 η dσ U 2σ ⊗ ∆U 2σ, ∇ U 4s−2σ
I 0+ I 0+ Z Z s − 2 η dσ U 2σ ⊗ U 2σ, ∇ ∆U 4s−2σ
I 0+

Note that for the third component, we used some properties from Fact 11 to move the Laplacian. It also explains our choice of W. We now use Penrose notation to estimate the 3 components. To clean up the notation, we only R R s focus on the integral on M, with the other integrals 2 I η 0+ dσ (·) in variables t and σ implicitly understood. We also use schematic identities for linear combinations of similar-looking tensor terms where we do not care how the indices contract (recall Equation (2)). By the error estimates above, all the terms with R or ν will be negligible as s ↓ 0, and interchanging derivatives will be a 4s−2σ free action. We write ≈ to throw the negligible error terms away. Also, when we write (∇jUl) , we mean the heat flow is applied to U, not ∇U (which is not possible anyway). First component: ( Z Z (((( Z 2σ 2σ 4s−2σ
2σ (2(σ ( 4s−2σ
i j
2σ l
2σ 4s−2σ ∆U ⊗ U , ∇ U = R ∗( U((∗ U ∗ ∇ U + ∇i∇ U U (∇jUl) M (M(( M ((( ((( Z 2σ (( Z ((( i j
2σ (l
(( 4s−2σ i j
2σ l(
2(σ 4s−2σ ≈ νi∇ U ((U (∇jUl) − ∇ U ((∇ (U (∇ U ) ((( ((( i j l (∂M(( (M(( Z i j
2σ l
2σ 4s−2σ − ∇ U U (∇i∇jUl) M Second component: ( Z Z (((( Z 2σ 2σ 4s−2σ
2σ (2(σ ( 4s−2σ
j
2σ i l
2σ 4s−2σ U ⊗ ∆U , ∇ U = U (∗(R(∗ U ∗ ∇ U + U ∇i∇ U (∇jUl) M (M(( M ( ( Z ((( Z (((( j 2σ i l (2σ(( 4s−2σ 2σ 2(σ ( 4s−2σ ≈ U
ν(∇(U(
(∇ U ) − ∇ U j
(∇i(U l(
(∇ U ) ((( i j l (i ((( j l (∂M(( (M(( Z j
2σ i l
2σ 4s−2σ − U ∇ U (∇i∇jUl) M

For the third component, note ∇ (R ∗ U) = ∇R ∗ U + R ∗ ∇U (( Z Z ((( Z 2σ 2σ 4s−2σ
2σ 2σ ((( 4s−2σ
j
2σ l
2σ i
4s−2σ − U ⊗ U , ∇ ∆U = − U (∗( U( ∗ ∇ R ∗ U − U U ∇j∇ ∇iUl M (M(( M Z ( j
2σ l
2σ (4(s−(2σ
i 4s−2σ
≈ − U U (R (∗ ∇( U + ∇ ∇j∇iUl M Z ( j
2σ l
2σ (4(s−(2σ
i 4s−2σ
≈ − U U (∇(R(∗ U + ∇ ∇i∇jUl M ((( Z 2σ 2σ ((( 4s−2σ Z ≈ − U j
U l
((ν(i∇ ∇ U
+ i j
2σ l
2σ 4s−2σ (((( i j l ∇ U U (∇i∇jUl) (∂M(( M

19 Z j
2σ i l
2σ 4s−2σ + U ∇ U (∇i∇jUl) M

R R s s↓0 Add them up, and we get 0 as 2 I η 0+ dσ (·) −−→ 0.

So we are done and the rest of the paper is to develop the tools we have borrowed for the proof.

4 Functional analysis

4.1 Common tools We note a useful inequality:

Theorem 20 (Ehrling’s inequality). Let X,Y, Xe be (real/complex) Banach spaces such that X is reflexive and X,→ Xe is a continuous injection. Let T : X → Y be a linear compact operator. Then ∀ε > 0, ∃Cε > 0:

kT xk ≤ ε kxk + C kxk ∀x ∈ X Y X ε Xe

Remark. Usually, X is some higher-regularity space than Xe (e.g. H1 and L2). The inequality is useful when the higher-regularity norm is expensive. We will need this for the Lp-analyticity of the heat flow (Theorem 71).

Proof. Proof by contradiction: Assume ε > 0 and there is (xj) such that kxjk = 1 and j∈N X kT x k > ε + j kx k . Since X is reflexive, by Banach-Alaoglu and PTAS, WLOG assume j Y j Xe X Y Xe xj −* x∞. Then T xj −* T x∞ and xj −* x∞. As T is compact, PTAS, WLOG T xj → T x∞. So kT x k ≥ lim sup ε + j kx k
> 0 and x −→Xe 0. Then x −X*e 0 and x = 0, contradict- ∞ Y j→∞ j Xe j j ∞ ing kT x∞kY > 0.

Definition 21 (Banach-valued holomorphic functions). Let Ω ⊂ C be an open set and X be a complex Banach space. Then a function f :Ω → X is said to be holomorphic (or analytic) when ∀z ∈ Ω: 0 f(z+h)−f(z) f (z) := lim|h|→0 h exists. The words “holomorphic” and “analytic” are mostly interchangeable, but “analytic” stresses the existence of power series expansion and can also describe functions on R for which analytic continuation into the complex plane exists.

Theorem 22 (Identity theorem). Let X be a complex Banach space and X0 ≤ X closed. Let Ω ⊂ C be connected, open and f :Ω → X holomorphic. Assume there is a sequence (zj) such that zj → z ∈ Ω and j∈N f(zj) ∈ X0 ∀j. Then f(Ω) ⊂ X0.

∗ Proof. Let Λ ∈ X such that Λ(X0) = 0. Reduce this to the scalar version in complex analysis.

In fact, many theorems from scalar complex analysis similarly carry over via linear functionals (cf. [Rud91, Theorem 3.31]).

4.2 Interpolation theory We will quickly review the theory of complex and real interpolation, and state the abstract Stein interpolation theorem. Interpolation theory can be seen as vast generalizations of the Marcinkiewicz and Riesz-Thorin interpolation theorems.

20 Definition 23. An interpolation couple of (real/complex) Banach spaces is a pair (X0,X1) of Banach spaces with a Hausdorff TVS X such that X0 ,→ X ,X1 ,→ X are continuous injections. Then X0 ∩ X1 and X0 + X1 are Banach spaces under the norms

kxk = max kxk , kxk
and kxk = inf kx k + kx k X0∩X1 X0 X1 X0+X1 0 X0 1 X1 x=x0+x1,xj ∈Xj

Let (Y0,Y1) be another interpolation couple. We say T :(X0,X1) → (Y0,Y1) is a morphism when T ∈ L (X0 + X1,Y0 + Y1) and T ∈ L (Xj,Yj) for j = 0, 1 under domain restriction. That implies T ∈ L (X0 ∩ X1,Y0 ∩ Y1) and we write T ∈ L ((X0,X1) , (Y0,Y1)). We also write L ((X0,X1)) = L ((X0,X1) , (X0,X1)).

2 Let P ∈ L((X0,X1)) such that P = P . Then we call P a projection on the interpolation couple (X0,X1).

Definition 24. Let (X0,X1) be an interpolation couple of (real/complex) Banach spaces. Then define the J-functional:

J : (0, ∞) × X0 ∩ X1 −→ R (t, x) 7−→ kxk + t kxk X0 X1 For θ ∈ (0, 1), q ∈ [1, ∞], define the real interpolation space   X −jθ j
q  (X0,X1) θ,q = uj : uj ∈ X0 ∩ X1, 2 J(2 , uj) ∈ l ( ) j∈Z j Z j∈Z  −jθ j P which is Banach under the norm kxk = inf 2 J(2 , uj) q . Note that uj denotes a (X0,X1)θ,q P l j∈ x= uj j Z j∈Z series that converges in X0 + X1. P • When q ∈ [1, ∞] and x ∈ X0 ∩ X1, note that ∀j ∈ : x = δkjx and Z k∈Z

−jθ j −jθ j(1−θ) 1−θ θ kxk ≤ inf 2 J(2 , x) = inf 2 kxk + 2 kxk ∼¬θ,¬q kxk kxk (X0,X1)θ,q X0 X1 X0 X1 j∈Z j∈Z

The last estimate comes from AM-GM and shifting j so that kxk ∼ 2j kxk . Note that the implied X0 X1 constants do not depend on θ and q.

• By considering the finite partial sums P u , we conclude that X ∩ X is dense in (X ,X ) |j|

• Let (Y0,Y1) be another interpolation couple and T ∈ L ((X0,X1) , (Y0,Y1)). For θ ∈ (0, 1), q ∈ [1, ∞],

define Xθ,q = (X0,X1)θ,q ,Yθ,q = (Y0,Y1)θ,q. Then T ∈ L(Xθ,q,Yθ,q) and

kT k kT k1−θ kT kθ L(Xθ,q ,Yθ,q ) .¬θ,¬q,¬T L(X0,Y0) L(X1,Y1)

where the implied constant does not depend on θ and q. This can be proved by a simple shifting argument.

• If P is a projection on (X0,X1) then (PX0,PX1)θ,q = P (X0,X1)θ,q.

Remark. There is also an equivalent characterization by the K-functional, which we shall omit. This theory can also be extended to quasi-Banach spaces. We refer to [BL76; Tri10] for more details.

21 Definition 25. Let (X0,X1) be an interpolation couple of complex Banach spaces. Let Ω = {z ∈ C : 0 < Re z < 1}. We then define the Banach space of vector-valued holomorphic/analytic functions on the strip:

|t|→∞ F = {f ∈ C0(Ω → X + X ): f holomorphic in Ω, kf(it)k + kf(1 + it)k −−−−→ 0} X0,X1 0 1 X0 X1
with the norm kfk = max supt∈ kf(it)k , supt∈ kf(1 + it)k . FX0,X1 R X0 R X1

For θ ∈ [0, 1], define the complex interpolation space [X0,X1]θ = {f(θ): f ∈ FX0,X1 }, which is Banach under the norm kxk = inf kfk [X0,X1]θ FX0,X1 f∈FX0,X1 f(θ)=x

2 2 • When x ∈ X ∩ X \{0} , θ ∈ [0, 1], ε > 0, define f (z) = eε(z −θ ) x . By the freedom in 0 1 ε kxk1−z kxkz X0 X1 choosing ε, we conclude

1−θ θ  ε(1−θ2) −θ2  1−θ θ 1−θ θ kxk[X ,X ] ≤ inf kfεkF kxkX kxkX ≤ inf max e , e kxkX kxkX = kxkX kxkX 0 1 θ ε>0 X0,X1 0 1 ε>0 0 1 0 1

• When θ ∈ [0, 1], by Poisson summation and Fourier series, we can prove that

N 2 X F 0 = {eCz eλj zx : N ∈ , C > 0, λ ∈ , x ∈ X ∩ X } X0,X1 j N j R j 0 1 j=1

is dense in FX0,X1 (cf. [BL76, Lemma 4.2.3]). This implies X0 ∩ X1 is dense in [X0,X1]θ. There is a simple extension of the above density result. Let U be dense in X0 ∩ X1 and define A(Ω) = {φ ∈ C0(Ω → C): φ holomorphic in Ω}. Then

N 2 X F U = {eCz φ (z)u : N ∈ , C > 0, φ ∈ A(Ω), u ∈ U} X0,X1 j j N j j j=1

is dense in FX0,X1 . This will lead to the abstract Stein interpolation theorem.

• Let (Y0,Y1) be another interpolation couple and T ∈ L ((X0,X1) , (Y0,Y1)). Then for θ ∈ [0, 1], almost by the definitions, we conclude

kT k ≤ kT k1−θ kT kθ L([X0,X1]θ ,[Y0,Y1]θ ) L(X0,Y0) L(X1,Y1)

• If P is a projection on (X0,X1) then [PX0,PX1] θ = P [X0,X1] θ

Remark. A keen reader would notice that we use square brackets for complex interpolation, and parentheses for real interpolation. One reason is that the real interpolation methods easily extend to quasi-Banach spaces, while the complex interpolation method does not. There is a version of complex interpolation for special quasi-Banach spaces, which is denoted by parentheses (cf. [Tri10, Section 2.4.4]), but we shall omit it for simplicity.

Blackbox 26 (Abstract Stein interpolation). Let (X0,X1) and (Y0,Y1) be interpolation couples of complex

Banach spaces and U dense in X0 ∩ X1. Let Ω = {z ∈ C : 0 < Re z < 1} and (T (z))z∈Ω be a family of linear mappings T (z): U → Y0 + Y1 such that
1. ∀u ∈ U : Ω → Y0 + Y1, z 7→ T (z)u is continuous, bounded and analytic in Ω.

22 2. For j = 0, 1 and u ∈ U: ( → Y , t 7→ T (j + it)u) is continuous and bounded by M kuk for some R j j Xj Mj > 0.

Then for θ ∈ [0, 1], we can conclude

kT (θ)uk ≤ M 1−θM θ kuk ∀u ∈ U [Y0,Y1]θ 0 1 [X0,X1]θ

Consequently, by unique extension, we have T (θ) ∈ L([X0,X1]θ , [Y0,Y1]θ).

Proof. See [Voi92], which is a very short read.

Remark. We will only use Stein interpolation in Subsection 4.3.

4.3 Stein extrapolation of analyticity of semigroups We are inspired by [Fac15, Theorem 3.1.1] (Stein extrapolation) and [Fac15, Theorem 3.1.10] (Kato-Beurling extrapolation), and wish to create variants for our own use. We will focus on Stein extrapolation, since it is simpler to deal with. There exists a subtle, but very important criterion to establish analyticity/holomorphicity:

Blackbox 27 (Holo on total). Let Ω ⊂ C be open and X complex Banach. Let f :Ω → X be a function. Assume N ≤ X∗ is total (separating points) and f is locally bounded. Then f is analytic iff Λf is analytic ∀Λ ∈ N.

Proof. This is a consequence of Krein-Smulian and the Vitali holomorphic convergence theorem, and we refer to [Are+11, Theorem A.7].

Remark. It will quickly become obvious how crucial this criterion is for the rest of the paper. Let us briefly note that an improvement has just been discovered by Arendt et al. [ABK19] (the author thanks Stephan Fackler for bringing this news).

Corollary 28 (Inheritance of analyticity). Let Ω ⊂ C be open and X,Y be complex Banach spaces where j : X,→ Y is a continuous injection. Let f :Ω → X be locally bounded. Then f is analytic iff j ◦ f is analytic.

Proof. Im(j∗) is weak∗-dense, therefore total.

Corollary 29 (Evaluation on dense set). Let X,Y be complex Banach spaces with X0 ≤ X. Let Ω ⊂ C be open and f :Ω → L(X,Y ) be a function. Assume X0 ≤ X is weakly dense and f is locally bounded. Then f is analytic ⇐⇒ ∀x0 ∈ X0, f(·)x0 :Ω → Y is analytic.

∗ ∗ ∗ ∗ Proof. Consider NX0 = span{y ◦ evx0 : x0 ∈ X0, y ∈ Y } ≤ L(X,Y ) . It is total as X0 is weakly dense. Use Blackbox 27.

23 4.3.1 Semigroup definitions As mentioned before, we assume the reader is familiar with basic elements of functional analysis, including semigroup theory as covered in [Tay11a, Appendix A.9]. Unfortunately, definitions vary depending on the authors, so we need to be careful about which ones we are using.

+ − + Definition 30. For δ ∈ (0, π], define Σδ = {z ∈ C\{0} : | arg z| < δ},Σδ = −Σδ , D = {z ∈ C : |z| < 1}. + − + Also define Σ0 = (0, ∞) and Σ0 = −Σ0 . Let X be a complex Banach space.

(T (t))t≥0 ⊂ L(X) is called:

• a semigroup when T : [0, ∞) → L(X) is a monoid homomorphism (T (0) = 1,T (t1 +t2) = T (t1)T (t2))

• degenerate when T : (0, ∞) → L(X) is continuous in the SOT (strong operator topology).

• immediately norm-continuous when T : (0, ∞) → L(X) is norm-continuous.

• C0 (strongly continuous) when T : [0, ∞) → L(X) is continuous in the SOT.

• bounded when T ([0, ∞)) is bounded in L(X), and locally bounded when T (K) is bounded ∀K ⊂

[0, ∞) bounded. (so C0 implies local boundedness by Banach-Steinhaus, and the semigroup property implies we just need to test K ⊂ [0, 1))

(T (z)) + ⊂ L(X) is called z∈Σδ ∪{0} + • a semigroup when T :Σδ ∪ {0} → L(X) is a monoid homomorphism.

0 + • C0 when ∀δ ∈ (0, δ),T :Σδ0 ∪ {0} → L(X) is continuous in the SOT.

+
0 • bounded when T Σδ0 is bounded ∀δ ∈ (0, δ) and locally bounded when T (K) is bounded ∀K ⊂ + Σδ0 bounded. (so C0 implies local boundedness, and the semigroup property implies we just need to + test K ⊂ D ∩ Σδ0 ) + • analytic when T :Σδ → L(X) is analytic

π We say (T (t))t≥0 is analytic of angle δ ∈ (0, ] if there is an extension (T (z)) + ⊂ L(X) which is 2 z∈Σδ ∪{0} analytic and locally bounded. If furthermore (T (z)) + is bounded, we say (T (t))t≥0 is boundedly z∈Σδ ∪{0} analytic of angle δ.

Remark. A subtle problem is that when (T (t))t≥0 is bounded and analytic, we cannot conclude (T (t))t≥0 is boundedly analytic (cf. [Are+11, Definition 3.7.3]).

π Blackbox 31. If (T (t))t≥0 is a C0 semigroup which is (boundedly) analytic of angle δ ∈ (0, 2 ], then (T (z)) + is a C0, (bounded) semigroup. z∈Σδ ∪{0}

Proof. The semigroup property comes from the identity theorem, and C0 comes from the Vitali holomorphic convergence theorem. We refer to [Are+11, Proposition 3.7.2].

tA Theorem 32 (Sobolev tower). Let (e )t≥0 be a C0 semigroup on a (real/complex) Banach space X with m generator A (implying A is closed and densely defined). Then ∀m ∈ N1, D(A ) is a Banach space under Pm k m the norm kxkD(Am) = kxkX + k=1 A x X , and D(A ) is dense in X.

24 tA tA
As e and A commute on D(A), we conclude that e t≥0, after domain restriction, is also a C0 m tA tA semigroup on D(A ) and e L(D(Am)) ≤ e L(X) ∀t ≥ 0. tA
tA
Lastly, if X is a complex Banach space and e t≥0 is (boundedly) analytic on X, e t≥0 is also (boundedly) analytic on D(Am) after domain restriction.

Proof. Most are just the basics of semigroup theory (cf. [Tay11a, Appendix A.9]). We only prove tA
zA the last assertion. All we need is commutativity: if e is extended to (e ) + , we want t≥0 z∈Σδ ∪{0} to show ezAA = AezA on D(A). zA + By Blackbox 31,(e ) + is a C0 semigroup. Therefore ∀x ∈ D(A), ∀z ∈ Σ : z∈Σδ ∪{0} δ

 etA − 1  etA − 1 etA − 1 ezAAx = ezA X- lim x = X- lim ezA x = X- lim ezAx t↓0 t t↓0 t t↓0 t

The last term implies ezAx ∈ D (A) and ezAAx = AezAx. Then use Corollary 28 and Corollary 29 to get analyticity.

4.3.2 Simple extrapolation (with core) Lemma 33. Let U, X be complex Banach spaces and U,→ X be a continuous injection with dense image.

1. Let (T (t))t≥0 ⊂ L(X) be locally bounded and T (t)U ≤ U ∀t ≥ 0. Assume (T (t))t≥0 is a C0 semigroup on U. Then (T (t))t≥0 on X is also a C0 semigroup.

π + 2. Let (T (z)) + ⊂ L(X) (where δ ∈ (0, ]) be locally bounded and T (z)U ≤ U ∀z ∈ Σ . Assume z∈Σδ ∪{0} 2 δ (T (z)) + is a C0, analytic semigroup on U. Then (T (z)) + on X is also a C0, analytic z∈Σδ ∪{0} z∈Σδ ∪{0} semigroup.

Remark. The assumption of local boundedness on X is important. We will also use this result in Subsec- tion 7.3 to establish the W 1,p-analyticity of the heat flow.

Proof. The semigroup property comes from the density of U in X.

To get C0 on X, use the local boundedness on X and dense convergence (Lemma9). For analyticity in (2), use Corollary 29.

Lemma 34 (Core). Let A be an unbounded linear operator on a (real/complex) Banach space X and E ≤   D(A). E is called a core when E is dense in D(A), k·kD(A) . tA If A is the generator of a C0 semigroup on X, E is dense in X and e E ≤ E, then E is a core.

X Proof. Let x ∈ D(A). Then there is (xj) in E such that xj −→ x. It is trivial to check j∈N

Z t k·k Z t k·k 1 sA D(A) 1 sA D(A) e xj ds −−−−−→ e x ds −−−−−→ x t 0 j→∞ t 0 t↓0

 sA  R t sA as e is k·kD(A)-continuous. Note that e xj ds is in the k·kD(A)-closure of E by the D(A) s≥0 0 Riemann integral.

25 Theorem 35 (Simple extrapolation with core). Let (X0,X1) be an interpolation couple of complex Banach spaces and Xθ = [X0,X1]θ for θ ∈ (0, 1].

Let (T (t))t≥0 ⊂ L ((X0,X1)) . Assume that on X0, (T (t))t≥0 is bounded. π Assume that on X1, (T (t))t≥0 is a C0 semigroup, boundedly analytic of angle δ ∈ (0, 2 ] with generator A1.  m 
Assume ∃m ∈ 1 : D(A ), k·k m ,→ X0 ∩ X1, k·k ,→ X0 are continuous injections with N 1 D(A1 ) X0∩X1 dense images.

Then on Xθ, (T (t))t≥0 is a C0 semigroup, and boundedly analytic of angle θδ.

m Remark. The existence of a convenient core like D(A1 ) is usually a trivial consequence of Sobolev embedding. We can replace bounded analyticity on X1 and Xθ with analyticity, and boundedness on X0 with local 0 π −C 0 z boundedness via the usual rescaling argument (∀δ ∈ (0, δ) ⊂ (0, ), ∃Cδ0 > 0 : e δ T (z) δ0 2 L(X1) . + 1 ∀z ∈ Σδ0 ). The existence of a core allows conditions on X0 and X1 to be more general than those in [Fac15, Theorem 3.1.1] (which requires immediate norm-continuity on X0), and actually be equivalent to those in [Fac15, Theorem 3.1.10] (though Kato-Beurling covers more than just complex interpolation). Once again, the assumption of (local) boundedness on X0 is important. We will use this result to establish the Lp-analyticity of the heat flow in Subsection 7.2.

m Proof. Let U = D(A1 ). Then U is Banach as A1 is closed. Obviously U,→ Xθ is a continuous injection with dense image, and (T (z)) + is a C0, bounded, analytic semigroup on U (via z∈Σδ ∪{0} Sobolev tower).

By Lemma 33,(T (t))t≥0 is a C0, bounded semigroup on X0. Also by Lemma 33, to get the desired conclusion, we only need to show (T (z)) + is locally bounded in L(Xθ). z∈Σθδ ∪{0} Fix δ0 ∈ (0, δ). We use abstract Stein interpolation. Define the strip Ω = {0 < Re < 1}. Let α ∈ (−δ0, δ0), ρ > 0, u ∈ U and L(z) = T (ρeiαz)u ∀z ∈ Ω

Note that U ≤ X0 ∩ X1 is dense. We check the other conditions for interpolation:

• As U,→ X0 and U,→ X1 are continuous, (Ω → X0 + X1, z 7→ L(z)u) is continuous, bounded on Ω and analytic on Ω (as L(z)u ∈ X1 ,→ X0 + X1).

• For j = 0, 1 (R → Xj, s 7→ L(j + is)u) is

– continuous since U,→ Xj is continuous. – bounded by C kuk for some C > 0 since (T (t)) is bounded on X and T (teiα)
j,T Xj j,T t≥0 0 t≥0 is bounded on X1.

iθα 0 0 + Then by Stein interpolation, we conclude {T (ρe ): ρ > 0, α ∈ (−δ , δ )} = T (Σθδ0 ) ⊂ L(Xθ) is bounded.

4.3.3 Coreless version There is an alternative version which we will not use, but is of independent interest:

Theorem 36 (Coreless extrapolation). Let (X0,X1) be an interpolation couple of complex Banach spaces and Xθ = [X0,X1]θ for θ ∈ (0, 1].

Let (T (t))t≥0 ⊂ L ((X0,X1)) be a semigroup. Assume that on X0, (T (t))t≥0 is bounded and degenerate.

26 π Assume that on X1, (T (t))t≥0 is a C0 semigroup, boundedly analytic of angle δ ∈ (0, 2 ] with generator A1.

Then on Xθ, (T (t))t≥0 is a C0 semigroup, boundedly analytic of angle θδ. Remark. The differences with the previous version are underlined. Again, via rescaling we can replace bounded analyticity on X1 and Xθ with analyticity, and boundedness on X0 with local boundedness. The conditions on X0 and X1 are still a bit more general than those in [Fac15, Theorem 3.1.1], which requires immediate norm-continuity on X0. In practice local boundedness on X0 can usually come from global analysis, while degeneracy can come from Sobolev embedding and dense convergence (Lemma9). Immediate norm-continuity is harder to establish. Note that Theorem 36 is not as general as [Fac15, Theorem 3.1.10] (which removes the need for degeneracy and covers more than just complex interpolation), though it is markedly easier to prove.

Proof. By interpolation, (T (t))t≥0 is a bounded semigroup on Xθ. Let U = X0 ∩ X1. Obviously (T (t))t≥0 is a bounded semigroup on U. Then observe that ∀u ∈ U, ∀t, t0 ≥ 0 :

k(T (t) − T (t )) uk ≤ k(T (t) − T (t )) uk1−θ k(T (t) − T (t )) ukθ k(T (t) − T (t )) ukθ 0 Xθ 0 X0 0 X1 . 0 X1

Xθ Since θ 6= 0, we have T (t)u −−−→ T (t0)u. As (T (t))t≥0 is bounded on Xθ and U is dense in Xθ, we t→t0 conclude (T (t))t≥0 is C0 on Xθ by dense convergence (Lemma9). Fix δ0 ∈ (0, δ). We use abstract Stein interpolation. Define the strip Ω = {0 < Re < 1}. Let α ∈ (−δ0, δ0), ρ > 0, u ∈ U and L(z) = T (ρeiαz)u ∀z ∈ Ω

Note that U = X0 ∩ X1. We check the other conditions for interpolation:

• As U,→ X0 and U,→ X1 are continuous, (Ω → X0 + X1, z 7→ L(z)u) is continuous, bounded on Ω and analytic on Ω (as L(z)u ∈ X1 ,→ X0 + X1).

• For j = 0, 1 (R → Xj, s 7→ L(j + is)u) is

iα
– continuous since (T (t))t≥0 is degenerate on X0 and T (te ) t≥0 is C0 on X1. – bounded by C kuk for some C > 0 since (T (t)) is bounded on X and T (teiα)
j,T Xj j,T t≥0 0 t≥0 is bounded on X1.

iθα 0 0 + By Stein interpolation, {T (ρe ): ρ > 0, α ∈ (−δ , δ )} = T (Σθδ0 ) ⊂ L(Xθ) is bounded. Finally, we just need to show (T (z)) + is analytic on Xθ. Let u ∈ U. Then z∈Σθδ ∪{0}

+
Σδ → X1 ,→ X0 + X1, z 7→ T (z)u

+
+
is analytic. Therefore Σθδ → Xθ ,→ X0 + X1, z 7→ T (z)u is analytic, while Σθδ → Xθ, z 7→ T (z)u +
is locally bounded, so we can use Corollary 28 to conclude Σθδ → Xθ, z 7→ T (z)u is analytic. As +
U is dense in Xθ, by corollary 29, we conclude Σθδ → L (Xθ) , z 7→ T (z) is analytic.

4.4 Sectorial operators

Recall that if (T (t))t≥0 is a C0 semigroup on a complex Banach space X, then it has a closed, densely tA tA Ct defined generator A, and T (t) = e is exponentially bounded: e .¬t e for some C > 0. Then

27 ∀ζ ∈ {Re > C} : ζ ∈ ρ(A) and

1 Z ∞ x = e−ζtetAx dt ∀x ∈ X ζ − A 0 (cf. [Tay11a, Appendix A, Proposition 9.2]) 1 tA This means that the resolvent ζ−A is the Laplace transform of the semigroup e . This naturally leads to the question when we can perform the inverse Laplace transform, to recover the semigroup from the resolvent. This motivates the definition of sectorial operators, which includes the Laplacian. Unfortunately, there are wildly different definitions currently in use by authors. The reader should study the definitions closely whenever they consult any literature on sectorial operators (e.g. [Lun95; Haa06; Are+11; Eng00]).

Definition 37. Let A be an unbounded operator on a complex Banach space X. For θ ∈ [0, π), we say A is  σ(A) ⊂ Σ−  θ • sectorial of angle θ (A ∈ Sect(θ)) when λ ∀ω ∈ [0, π − θ): M(A, ω) := sup λ−A < ∞  + λ∈Σω

• quasi-sectorial when ∃a ∈ R : A − a is sectorial.

π • acutely sectorial when A ∈ Sect(θ) for some θ ∈ [0, 2 )

• acutely quasi-sectorial when ∃a ∈ R : A − a is acutely sectorial.

π For r > 0, η ∈ ( 2 , π), we define the (counterclockwise-oriented) Mellin curve

iη −iη i[−η,η] γr,η = e [r, ∞) ∪ e [r, ∞) ∪ re

Remark. Depending on the author, “sectorial” can mean any of those four, and that is not taking sign conventions into account (some authors want −∆ to be sectorial), as well as whether A should be densely defined. The term “quasi-sectorial” is taken from [Haa06]. In particular, letting the spectrum be in the left half-plane means we agree with [Eng00; Lun95] and disagree with [Are+11; McI86; Haa06]. This is simply a personal preference, of being able to say “the

Laplacian is sectorial”, or “generators of C0 analytic semigroups are acutely sectorial”. Also, for bounded t∆ tz holomorphic calculus, e morally comes from (e )z∈σ(∆) which is bounded in the left half-plane. In keeping with tradition, here is the usual visualization:

28 Figure 1: Acutely sectorial operators

Blackbox 38. A generates a C0, boundedly analytic semigroup on complex Banach space X if and only if A is densely defined and acutely sectorial. π
π tA
ζA
When that happens, ∃δ ∈ 0, 2 and η ∈ ( 2 , π) such that e extends to e + and t≥0 ζ∈Σδ ∪{0} Z ζA 1 ζz 1 + e = e dz ∀ζ ∈ Σδ , ∀r > 0 2πi γr,η z − A

tA ∞ k tA kk k tA k tA Also ∀t > 0, ∀k ∈ N1 : e (X) ≤ D(A ), A e .¬t,¬k tk and ∂t (e x) = A e x ∀x ∈ X. Remark. This is the aforementioned inverse Laplace transform. The Mellin curve and the resolvent estimate in the definition of sectoriality ensure sufficient decay for the integral to make sense. As it is a complex line integral and the resolvent is analytic, the semigroup becomes analytic. A trivial consequence is that D(A∞) is dense in X and therefore a core. When A is densely defined and acutely quasi-sectorial, a simple rescaling et(A−a) = e−taetA implies tA
e t≥0 is a C0, analytic semigroup.

k  t k k k tA k A Proof. See [Eng00, Section II.4.a]. The curious figure tk comes from A e = Ae .

Theorem 39 (Yosida’s half-plane criterion). A is acutely quasi-sectorial if and only if ∃C > 0 such that

•{ Re > C} ⊂ ρ(A)

• sup λ < ∞ λ−A λ∈{Re>C} Remark. This is how the Lp-analyticity of the heat flow is traditionally established. Yet proving the resolvent estimate is nontrivial, as it is quite a refinement of elliptic estimates, so we choose not to do so. Interestingly, 1 3 we will instead use this for the B3,1-analyticity of the heat flow in Subsection 8.3, though that case is especially easy since we already have analyticity at the two endpoints L3 and W 1,3.

29  −1 Proof. We only need to prove ⇐. Recall the proof of how ρ(A) is open: ∀λ ∈ ρ(A),B λ, 1 ⊂ λ−A + π ρ(A). Applying this allows us to open up {Re > C} and get C + Ση ⊂ ρ(A) for some η ∈ ( 2 , π). π By choosing η near 2 , the resolvent estimate is retained.

Definition 40. Let A be an unbounded operator on a Hilbert space X. Then A is called

• symmetric when hAx, yi = hx, Ayi ∀x, y ∈ D(A), or equivalently, A ⊂ A∗ (where A and A∗ are identified with their graphs).

• self-adjoint when A = A∗. This implies σ(A) ⊂ R (cf. [Tay11a, Appendix A, Proposition 8.5]). • dissipative when Re hAx, xi ≤ 0 ∀x ∈ D(A).

When A is dissipative, ∀λ ∈ {Re > 0}, ∀x ∈ D(A) : Re h(λ − A) x, xi ≥ Re hλx, xi so k(λ − A) xk ≥ Re λ kxk.  −1 Recall how ρ(A) is proved to be open: ∀λ ∈ ρ(A),B λ, 1 ⊂ ρ(A). Consequently, if A is λ−A

dissipative and ∃λ0 ∈ {Re > 0} ∩ ρ(A), we can conclude {Re > 0} ⊂ ρ(A).

Theorem 41 (Dissipative sectoriality). Assume X is a complex Hilbert space and A is an unbounded, self-adjoint, dissipative operator on X. Then A is acutely sectorial of angle 0.

Remark. Though standard, this might be the most elegant theorem in the theory, and later on will instantly imply the L2-analyticity of the heat flow in Subsection 7.1. The theorem can also be proved by Euclidean geometry. When X is separable, we can also use the spectral theorem for unbounded operators.

Proof. As A is self-adjoint, C\R ⊂ ρ(A). By dissipativity, we conclude σ(A) ⊂ (−∞, 0]. Also by self-adjointness, Re hAx, xi = hAx, xi ≤ 0 ∀x ∈ D(A).

Arbitrarily pick θ ∈ ( π , π). We want to show z 1 ∀z ∈ Σ+. 2 z−A .θ θ 1 2 1 Let x ∈ X and u = z−A x. As |hu, xi| ≤ kukX kxkX , we want to show kukX .θ z hu, xi . Note that 1 1 1 hu, xi = hu, (z − A)ui = hu, ui − hAu, ui z z z 1 WLOG assume kukX = 1. Then we want 1 .θ 1 − z hAu, ui . Note that − hAu, ui ≥ 0 and 1 + − z hAu, ui ∈ Σθ . Then we are done since

1 + 1 − hAu, ui ≥ dist(0, 1 + Σ ) > 0. z θ

+ By Euclidean geometry, we can even calculate dist(0, 1 + Σθ ). We will not need it though.

5 Scalar function spaces

Throughout this section, we work with complex-valued functions.

5.1 On Rn Definition 42. Here we recall the various (inhomogeneous) function spaces which are particularly suitable for interpolation. They are defined as subspaces of S0(Rn) with certain norms being finite:

30 Pm k k p 1. Lebesgue-Sobolev spaces: for m ∈ 0, p ∈ [1, ∞]: kfk m,p n ∼ ∇ f where ∇ f ∈ L N W (R ) k=0 p are tensors defined by distributions. It is customary to write Hm for W m,2.

s s s 2 2. Bessel potential spaces: for s ∈ , p ∈ [1, ∞]: kfk s,p n ∼ kh∇i fk where h∇i = (1 − ∆) is R H (R ) p the Bessel potential.

s 3. Besov spaces: for s ∈ , p ∈ [1, ∞], q ∈ [1, ∞]: kfk s n ∼ kP≤1fk + N kPN fk where R Bp,q (R ) p p q lN>1 PN and P≤N (for N ∈ 2Z) are the standard Littlewood-Paley projections (cf. [Tao06, Appendix A]).

s 4. Triebel-Lizorkin spaces: for s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞]: kfkF s ( n) ∼ kP≤1fkp+ N kPN fklq p,q R N>1 p Remark. As there are multiple characterizations for the same spaces, we only define up to equivalent norms. Of course, the topologies induced by equivalent norms are the same. s,p s In the literature, “Fractional Sobolev spaces” like W could either refer to Bp,p (Sobolev–Slobodeckij s,p s spaces) or H . We shall avoid using the term at all. There are also some delicate issues with F∞,q which we do not need to discuss here (cf. [Tri10, Section 2.3.4]).

Blackbox 43. Recall from harmonic analysis (cf.[Tri10, Section 2.5.6, 2.3.3, 2.11.2] and [Lem02, Part 1, Chapter 3.1]):

m,p n m,p n • W (R ) = H (R ) for m ∈ N0, p ∈ (1, ∞).

s n s,p n • Fp,2(R ) = H (R ) for s ∈ R, p ∈ (1, ∞).

m n m,p n m n • Bp,1(R ) ,→ W (R ) ,→ Bp,∞(R ) for m ∈ N0, p ∈ [1, ∞].

n m,p n s n s n •S (R ) is dense in W (R ), Bp,q(R ) and Fp,q(R ) for m ∈ N0, s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞).

s n s n s n • Bp,min(p,q)(R ) ,→ Fp,q(R ) ,→ Bp,max(p,q)(R ) for s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞].

s n
∗ −s n • Bp,q (R ) = Bp0,q0 (R ) for s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞). s n
∗ −s n Fp,q (R ) = Fp0,q0 (R ) for s ∈ R, p ∈ (1, ∞), q ∈ (1, ∞).

5.2 On domains

Definition 44. A C∞ domain Ω in Rn is defined as an open subset of Rn with smooth boundary, and scalar function spaces are then defined on Ω. If Ω ⊂ S ⊂ Ω, let function spaces on S implicitly refer to n n function spaces on Ω. This will make it possible to discuss function spaces on, for example, R+ ∩ BR (0, 1), or compact Riemannian manifolds with boundary.

Obviously, Sobolev spaces are still defined on domains by distributions. The big question is finding a s s good characterization for Bp,q and Fp,q on domains, when the Fourier transform is no longer available. This is among the main topics of Triebel’s seminal books. Let us review the results:

n n ∞ n Definition 45. Let Ω be either R , or the half-space R+, or a bounded C domain in R . s s s n s n Then Bp,q(Ω) and Fp,q(Ω) can simply be defined as the restrictions of Bp,q(R ) and Fp,q(R ) to Ω and

s n kfk s = inf{kF k s n : F ∈ B ( ),F | = f} for s ∈ ; p, q ∈ [1, ∞] Bp,q (Ω) Bp,q (R ) p,q R Ω R

s n kfk s = inf{kF k s n : F ∈ F ( ),F | = f} for s ∈ , p ∈ [1, ∞), q ∈ [1, ∞] Fp,q (Ω) Fp,q (R ) p,q R Ω R

31 A more useful characterization is via BMD (ball mean difference). Let τhf(x) = f(x+h) be the translation m m operator and ∆hf = τhf − f be the difference operator. Then for m ∈ N1, we can define ∆h = (∆h) as the m-th difference operator. As we need to stay on the domain Ω, define

1 V m(x, t) = (B(x, mt) ∩ Ω − x) for x ∈ Ω, t > 0, m ∈ m N1

m m l m So V (x, t) ⊂ B(0, t), x + mV (x, t) ⊂ Ω and ∆hf(x) is well-defined when h ∈ V (x, t). Also note for m n t ∈ (0, 1): |V (x, t)| ∼Ω,m t . Then by [Tri92, Section 3.5.3, 5.2.2]:

1. For m ∈ N1, p ∈ [1, ∞], q ∈ [1, ∞], s ∈ (0, m), r ∈ [1, p]:

−s m kfk s ∼ kfk + t k∆ f(x)k r 1 m (7) Bp,q (Ω) p h L ( n dh,V (x,t)) p h t Lx(Ω) q 1 L ( t dt,(0,1))

We carefully note here that m > s (the difference operator must be strictly higher-order than the regularity), and that the variable t is small, which will play a big role in Theorem 54. We also note that this is different from the classical characterization via differences ([Tri10, Section 3.4.2], [Tri92, Section 1.10.3]) which analysts might be more familiar with:

−s m kfk s ∼ kfk + |h| ∆h,Ωf(x) p Bp,q (Ω) p Lx(Ω) q dh L ( |h|n ,B(0,1))

m m where ∆h,Ωf(x) is the same as ∆h f(x), but zero wherever undefined, and m ∈ N1, p ∈ (1, ∞), q ∈ [1, ∞], s ∈ (0, m).

2. For m ∈ N1, p ∈ [1, ∞), q ∈ [1, ∞], s ∈ (0, m), r ∈ [1, p]:

−s m kfk s ∼ kfk + t k∆ f(x)k r dh m Fp,q (Ω) p h L ( n ,V (x,t)) q dt h t L ( ,(0,1)) p t Lx(Ω)

Blackbox 46 (Diffeomorphisms and smooth multipliers). Every diffeomorphism on Rn preserves (under pullback) the topology of

k,p n • W (R ) for k ∈ N0, p ∈ [1, ∞]

s n • Bp,q(R ) for s ∈ R, p ∈ [1, ∞], q ∈ [1, ∞]

s n • Fp,q(R ) for s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞]

∞ n Also on the same spaces, for φ ∈ Cc (R ), f 7→ φf is a bounded linear map . Remark. This allows us to trivially define function spaces on compact Riemannian manifolds with boundary via partitions of unity and give them unique topologies.

k,p s s Proof. For W it is trivial. For Bp,q and Fp,q, see [Tri92, Section 4.3, 4.2.2] and [Tri10, Section 2.8.2].

n ∞ n Blackbox 47 (Extension and trace). Let Ω be either the half-space R+ or a bounded C domain in R . 1. Stein extension: There exists a common (continuous linear) extension operator E : W k,p(Ω) ,→ k,p n W (R ) for all k ∈ N0, p ∈ [1, ∞]

32 2. Triebel extension: For any N ∈ N1, there exists a common (continuous linear) extension operator EN such that

N s s n (a) E : Bp,q(Ω) ,→ Bp,q(R ) for all |s| < N, p ∈ [1, ∞], q ∈ [1, ∞] N s s n (b) E : Fp,q(Ω) ,→ Fp,q(R ) for all |s| < N, p ∈ [1, ∞), q ∈ [1, ∞] 3. Trace theorems: Let n ≥ 2.

s− 1 1 s p (a) For p ∈ [1, ∞], q ∈ [1, ∞], s > p : Bp,q(Ω)  Bp,q (∂Ω) is a retraction (continuous surjection with a bounded linear section as a right inverse). s− 1 1 s p (b) For p ∈ [1, ∞), q ∈ [1, ∞], s > p : Fp,q(Ω)  Bp,p (∂Ω) is a retraction. 1 p p 1,1 1 (c) (Limiting case) For p ∈ [1, ∞),Bp,1(Ω)  L (∂Ω) and W (Ω)  L (∂Ω) are continuous surjections.

1 3 Remark. It is important to note that we do not have the trace theorem for, say, B3,2(Ω) (cf. [Sch11, Section 3])

Proof.

1. See [Ste71, Section VI.3].

2. See [Tri92, Section 4.5, 5.1.3].

3. See [Tri10, Section 2.7.2, 3.3.3] and the remarks.

n ∞ n Corollary 48. Let Ω be either the half-space R+ or a bounded C domain in R .

m m,p • Fp,2(Ω) = W (Ω) for m ∈ N0, p ∈ (1, ∞).

m m,p m • Bp,1(Ω) ,→ W (Ω) ,→ Bp,∞(Ω) for m ∈ N0, p ∈ [1, ∞].

m,p s s •S (Ω) is dense in W (Ω), Fp,q(Ω) and Bp,q(Ω) for m ∈ N0, s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞).

s s s • Bp,min(p,q)(Ω) ,→ Fp,q(Ω) ,→ Bp,max(p,q)(Ω) for s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞] Remark. When Ω is a bounded C∞ domain, S(Ω) = C∞(Ω).

Proof. Use Triebel and Stein extensions.

5.3 Holder & Zygmund spaces

n n ∞ n ∞ Definition 49. Let Ω be either R , the half-space R+ or a bounded C domain in R . Recall some L type spaces:

• Holder spaces: for k ∈ N0, α ∈ (0, 1],

 β  kfkCk,α(Ω) = kfkCk(Ω) + max D f 0,α |β|=k C (Ω)

|g(x)−g(y)| where [g]C0,α(Ω) = supx6=y |x−y|α

33 s s • Zygmund spaces: for s > 0, define C (Ω) = B∞,∞(Ω) . Then for m ∈ N1, s ∈ (0, m):

∆m f(x) −s m h,Ω kfk s ∼ kfk ∞ + t k∆h f(x)k ∞ m ∼ sup |f|+ sup C (Ω) L (Ω) Lh (V (x,t)) ∞ s Lx (Ω) ∞ 0<|h|≤1,x∈Ω |h| Lt ((0,1))

It is well-known (cf. [Tri10, Section 2.2.2, 2.5.7, 2.5.12, 2.8.3]) that

β •k fkCk+α(Ω) ∼ kfkCk + max|β|=k D f Cα(Ω) for k ∈ N0, α ∈ (0, 1]

•k fkCk+α(Ω) ∼ kfkCk,α for k ∈ N0, α ∈ (0, 1).

•k fgkCs(Ω) . kfkCs kgkCs for s > 0. Note that C0,1,C1 and C1 are different.

5.4 Interpolation & embedding

n n ∞ n Blackbox 50 (Interpolation). Let Ω be either R , the half-space R+ or a bounded C domain in R . Throughout the theorem, always assume θ ∈ (0, 1), sθ = (1 − θ)s0 + θs1.

1. Bs0 (Ω),Bs1 (Ω)
= Bsθ (Ω) for s 6= s , s ∈ , p ∈ [1, ∞], q , q ∈ [1, ∞]. p,q0 p,q1 θ,q p,q 0 1 j R j

F s0 (Ω),F s1 (Ω)
= Bsθ (Ω) for s 6= s , s ∈ , p ∈ [1, ∞), q , q ∈ [1, ∞]. p,q0 p,q1 θ,q p,q 0 1 j R j

s0 s1
sθ 1 1−θ θ 2. Bp ,q (Ω),Bp ,q (Ω) = Bp ,p (Ω) for s0 6= s1, sj ∈ , pj ∈ [1, ∞], qj ∈ [1, ∞], = + = 0 0 1 1 θ,pθ θ θ R pθ p0 p1 1−θ + θ q0 q1

3. Bs0 (Ω),Bs1 (Ω) = Bsθ (Ω) and F s0 (Ω),F s1 (Ω) = F sθ (Ω) p0,q0 p1,q1 θ pθ ,qθ p0,q0 p1,q1 θ pθ ,qθ

1 1−θ θ 1 1−θ θ for sj ∈ , pj ∈ (1, ∞), qj ∈ (1, ∞), = + , = + . R pθ p0 p1 qθ q0 q1

p0 p1 p 1 1−θ θ 4. [L (Ω),L (Ω)] = L θ (Ω) for pj ∈ [1, ∞] , = + . θ pθ p0 p1

m0,p m1,p mθ 5. (W (Ω),W (Ω))θ,q = Bp,q (Ω) for mj ∈ N0, m0 6= m1, p ∈ [1, ∞], q ∈ [1, ∞], mθ = (1 − θ) m0 + θm1.

Proof.

1. Extension operators and [Tri10, Section 2.4.2].

2. Extension operators and [BL76, Theorem 6.4.5].

3. Extension operators and [Tri10, Section 2.4.7].

4. Extension by zero and [BL76, Section 5.1.1]

m m,p m 5. Recall Bp,1(Ω) ,→ W (Ω) ,→ Bp,∞(Ω) for m ∈ N0, p ∈ [1, ∞]. Then apply 1.

∞ n Blackbox 51 (Embedding). Let Ω be a bounded C domain in R . Assume ∞ > s0 > s1 > −∞. Then

34 s0 s1 1 1 s0−s1 1. B (Ω) ,→ B (Ω) is compact for pj ∈ [1, ∞], qj ∈ [1, ∞], > − p0,q0 p1,q1 p1 p0 n

s0 s1 1 1 s0−s1 F (Ω) ,→ F (Ω) is compact for pj ∈ [1, ∞), qj ∈ [1, ∞], > − p0,q0 p1,q1 p1 p0 n

s0 s1 1 1 s0−s1 2. B (Ω) ,→ B (Ω) is continuous for pj ∈ [1, ∞], q ∈ [1, ∞], = − p0,q p1,q p1 p0 n

s0 s1 1 1 s0−s1 F (Ω) ,→ F (Ω) is continuous for pj ∈ [1, ∞), qj ∈ [1, ∞], = − p0,q0 p1,q1 p1 p0 n

Proof.

1. See [Tri10, Section 4.3.2, Remark 1] and [Tri10, Section 3.3.1].

2. See [Tri10, Section 3.3.1].

Corollary 52. Let Ω be a bounded C∞ domain in Rn. Then

1 1 m0−m1 1. For mj ∈ 0, m0 > m1, pj ∈ [1, ∞], > − : N p1 p0 n W m0,p0 (Ω) ,→ Bm0 (Ω) ,→ Bm1 (Ω) ,→ W m1,p1 (Ω) is compact. p0,∞ p1,1

1 m0−(m1+α) 2. For mj ∈ 0, m0 > m1, p0 ∈ [1, ∞], α ∈ (0, 1) , 0 > − : N p0 n W m0,p0 (Ω) ,→ Bm0 (Ω) ,→ Bm1+α(Ω) = Cm1,α(Ω) is compact. p0,∞ ∞,∞

1 m,p m p p 3. For m ∈ N1, p ∈ (1, ∞): W (Ω) ,→ Bp,∞(Ω) ,→ Bp,1(Ω)  L (∂Ω) is compact. Remark. These include the Rellich-Kondrachov embeddings found in [Ada03, Theorem 6.3], so the Besov embeddings generalize Sobolev embeddings.

5.5 Strip decay

Some notation first: let Ω be a C∞ domain in Rn or a compact Riemannian manifold with or without boundary. Define Ω>r = {x ∈ Ω : dist(x, ∂Ω) > r} where dist(x, ∂Ω) = ∞ if ∂Ω = ∅. Similarly define

Ω≥r, Ω

1 1 Z  p Z  p p 1 p kfk p = kf(x)k p dx = − |f| = |f| L (Ω,avg) Lx( ,Ω) 1/p |Ω| Ω |Ω| Ω

By convention, we set kfk ∞ = kf(x)k ∞ dx = kfk ∞ . The implicit measure is of course the L (Ω,avg) Lx ( |Ω| ,Ω) L (Ω) Riemannian measure. In such mean integrals, the domain becomes a probability space.

Theorem 53 (Coarea formula).

n m,p n s n 1. For any h ∈ R , the translation semigroup (τth)t≥0 is a C0 semigroup on W (R ), Bp,q(R ) and 1 s n p n Fp,q(R ) for m ∈ N0, s ∈ R, p ∈ [1, ∞), q ∈ [1, ∞). Consequently, for p ∈ [1, ∞) and f ∈ Bp,1(R ) ,

p n−1
[0, ∞) → L ( ), t 7→ τthf| n−1 R R

is continuous and bounded by C kfk 1 for some C > 0. p n Bp,1(R )

35 2. Let Ω be a bounded C∞ domain in Rn (or a compact Riemannian manifold with boundary). Let 1 p p ∈ [1, ∞). Then for f ∈ Bp,1(Ω) and r > 0 small:   (a) [0, r) → , ρ 7→ kfk p is continuous and bounded by C kfk 1 for some C > 0. R L (∂Ω>ρ) p Bp,1(Ω)

(b) kfk p ∼¬r kfk p L (Ω≤r ) L (∂Ω>ρ) p Lρ((0,r)) r↓0 (c) kfk p kfk 1 and kfk p −−→kfk p . L (Ω≤r ,avg) .¬r p L (Ω≤r ,avg) L (∂Ω,avg) Bp,1(Ω) 1 p p r↓0 (d) Let I ⊂ be an open interval and f ∈ L (I → B (Ω)),then kfk 1 ¬r kfk p p −−→ R p,1 p p & Lt L (Ω≤r ,avg) Lt Bp,1(Ω)

kfk p p . Lt L (∂Ω,avg)

1 1,p p 3. Let p ∈ [1, ∞), f ∈ W (Ω), show that kfk p r kfk 1,p + r kfk p for r > 0 small. L (Ω

Proof.

1. Use the density of S(Rn) and Lemma9. 2.

(a) By partition of unity, geodesic normals, diffeomorphisms and the smallness of r, reduce the problem to the half-space case, which is just 1). 1 p ∞ (b) Approximate f in Bp,1 by C (Ω) functions. This is the well-known coarea formula, which

corresponds to Fubini’s theorem in the half-space case. Note that kfk p is defined L (∂Ω>ρ) by the trace theorem. See [Cha06, Section III.5] for more details.

(c) For r small, |Ωr| ∼ |∂Ω|, so

kfk p ∼ kfk p ≤ sup kfk p L (Ω≤r ,avg) L (∂Ω>ρ,avg) p L (∂Ω>ρ,avg) Lρ((0,r),avg) ρ

r↓0 and kfk p −−→kfk p by continuity in a). L (∂Ω>ρ,avg) p L (∂Ω,avg) Lρ((0,r),avg) (d) Dominated convergence.

3. By the trace theorem, WLOG f ∈ C∞(Ω). By partition of unity and diffeomorphisms, WLOG n n−1 Ω = R+ = {(x, y): x ∈ R , y ≥ 0}. Then

kfk p ∼ kf(x, y)k p k∂yf(x, ρ)k 1 + |f(x, 0)| L (Ω

1 1 0 k∂yf(x, ρ)k p |y| p + r p kf(x, 0)k p . Lρ([0,y]) p Lx Ly ([0,r]) p Lx

1 p0 The first term k∂yf(x, ρ)k p |y| r k∂yfk p . So we are done. . Lρ([0,r]) p . L (Ω

Theorem 54 (Product estimate). Let M be a bounded C∞ domain in Rn (or a compact Riemannian ∞ manifold with boundary). Assume r > 0 small, fr ∈ C (M) with support in M

36 1 p g ∈ Bp,1(M):

1/p 1/p p ∞ 1/p kfrgk M,¬r kfrk kgkL (M<4r ) + kfrkL (M

p p application, fr has very small support and we want to use kgkL (M<4r ) instead of kgkL (M) to control the product. Unfortunately there does not seem to be much, if at all, literature on this issue. This theorem will only be used for Theorem 86, and is not necessary for Onsager’s conjecture.

n Proof. By diffeomorphisms, partition of unity, and geodesic normals, WLOG assume M = R+ with n M

p 1/p p p n Recall kgkL (xn=a) . kgk n ∀0 ≤ a < ∞ where kgkL (xn=a) := kgkL ({x∈ :xn=a}) is defined Bp,1 (R+) R by the trace theorem.

WLOG, assume kfrk∞ ≤ 1. Recall the characterization of Besov spaces by ball mean difference (BMD) and write V (x, t) for V 1(x, t) (see Equation (7)). Then

− 1 −n kfrgk 1/p ∼ kfrgkLp(M) + t p k∆h(frg)(x)k 1 B (M) Lh(V (x,t)) p p,1 Lx(M) 1 dt Lt ( t ,(0,1))

The term kfrgkLp(M) is easily bounded and thrown away. For the remaining term, we use the identity ∆h(frg) = ∆hfrg + τhfr∆hg to bound it by

− 1 −n − 1 −n t p k∆hfr(x)k 1 g(x) + t p kfrk k∆hg(x)k 1 Lh(V (x,t)) p ∞ Lh(V (x,t)) p Lx(M) 1 dt Lx(M) 1 dt L ( t ,(0,1)) L ( t ,(0,1))

The second term here is just kfrkL∞ kgk 1/p , so throw it away. For the remaining term, by using Bp,1 (M)

k·k p k·k p + k·k p and L (M) . L (M<4r ) L (M>4r )

k∆hfr(x)kL1 (V (x,t))g(x) k∆hfr(x)kL1 (V (x,t)) kg(x)k p h p . h ∞ Lx(M<4r ) Lx(M<4r ) Lx (M<4r ) we are left with

− 1 −n kfrk 1/p kgkLp(M ) + t p k∆hfr(x)k 1 g(x) B (M) <4r Lh(V (x,t)) p ∞,1 Lx(M>4r ) 1 dt L ( t ,(0,1))

Throwing away the first term, we have arrived at the important estimate: what happens on M>4r.

It will turn out that the values of g on M>4r are well-controlled by kgk 1/p . To begin, recall fr Bp,1 (M) is supported on M

−n −n |B(x, t) ∩ Mxn−r ∀x ∈ M>4r, ∀t ∈ (3r, 1) h h |B(x, t)| xn

Note that t > 3r comes from t > xn − r > 4r − r. So we have used the “room” from 4r to get an O(r)-lower bound for t. By xn < r + t, we now only need to bound

− 1 r t p g(x) xn p Lx(M[4r,r+t]) 1 dt L ( t ,(3r,1))

37 Obviously, we will integrate g on xn-slices (using p > 1):

r 1 1 1 p g(x) = r kgk p kgk 1/p r r kgk 1/p L (xn=ρ) . B (M) . B (M) xn p ρ p p,1 ρ p p,1 Lx(M[4r,r+t]) Lρ([4r,r+t]) Lρ([4r,∞))

Then we are done (using p < ∞):

1 1   p 1 − 1 r  p 1 r p t p = = ¬r 1 1 dt . L ( t ,(3r,1)) t 1 dt t L ( t ,(3r,1)) 1 dt 1 L ( t ,(3, r ))

6 Hodge theory

We stick closely to the terminology and symbols of [Sch95], with some careful exceptions.

6.1 The setting Definition 55. Define a ∂-manifold as a paracompact, Hausdorff, metric-complete, oriented, smooth manifold, with no or smooth boundary.

n n Note that this means BR (0, 1) is not a ∂-manifold (as it is not complete), but BR (0, 1) is. For the rest of this paper, unless mentioned otherwise, we work on M which is a compact Riemannian n-dimensional ∂-manifold (where n ≥ 2), and use ν to denote the outwards unit normal vector field on ∂M.

As before, define M>r = {x ∈ M : dist(x, ∂M) > r}, and similarly for M≥r,M 0 small, the map (∂M × [0, r) → M

Let vol stand for the Riemannian volume form orienting M and vol∂ for that of ∂M. Let  : ∂M ,→ M be the smooth inclusion map and ι stand for interior product (contraction) of differential forms. Note that ∗ for a smooth differential form ω,  ω only depends on ω ∂M , so by abuse of notation, we can write

∗ vol∂ =  (ιν vol)

n−1 R R ∗ n−1 where ιν vol ∈ Ω (M) ∂M . Additionally, the Stokes theorem reads M dω = ∂M  ω for ω ∈ Ω (M).

6.1.1 Vector bundles Let be a real vector bundle over M with a Riemannian fiber metric h·, ·i . F F Define

• Γ(F) : the space of smooth sections of F

• Γc(F) : smooth sections with compact support (so Γc(F) = Γ(F) since M is compact) ◦ • Γ00(F) : smooth sections with compact support in M (the interior of M). Remark. We are following [Sch95], where Hodge theory is also formulated for non-compact M. In the book, ◦ Γ0F is used instead of Γ00F to denote compact support in M. As that can be confused with having zero trace, we opt to write Γ00F instead.

38 Then on Γc(F), define the dot product Z hhσ, θii = hσ, θi vol F M

p p and |σ| = hσ, σi . Then for p ∈ [1, ∞), L Γ( ) is the completion of Γc( ) under the norm F F F F

kσk p = k|σ| k L Γ(F) F Lp(M)

Let ∇F be a connection on F. Then for σ ∈ Γ(F), ∇Fσ ∈ Γ(T ∗M ⊗ F) and we can define the fiber metric

hα ⊗ σ, β ⊗ θi ∗ = hα, βi ∗ hσ, θi T M⊗F T M F In local coordinates (Einstein notation):

F F i F j F i j F F ij F F ∇ σ, ∇ θ ∗ = dx ⊗ ∇i σ, dx ⊗ ∇j σ = dx , dx ∗ ∇i σ, ∇j θ = g ∇i σ, ∇j θ T M⊗F F T M F F For higher derivatives, define the k-jet fiber metric

D (j) (j) E X F
F
hσ, θiJ k = ∇ σ, ∇ θ F (Nj T ∗M)⊗ 0≤j≤k F p and |σ| k = hσ, σi k . Then we have Cauchy-Schwarz: |hσ, θi k | ≤ |σ| k |θ| k . J F J F J F J F J F m,p Then for m ∈ N0, p ∈ [1, ∞), we define the Sobolev space W Γ(F) as the completion of Γc(F) under the norm

kσk m,p = k|σ| m k W Γ(F) J F Lp(M)

It is worth noting that |σ| m , up to some constants, does not depend on ∇F. Indeed, assume there is J F another connection ∇e F, then ∇F − ∇e F is tensorial:       F F F F F F ∞ ∇X − ∇e X (fσ) = f ∇X − ∇e X (σ) = ∇fX − ∇e fX (σ) for f ∈ C (M), σ ∈ Γ(F),X ∈ XM

  ∞ F F So there is a C (M)-multilinear map A : XM ⊗C∞(M) Γ(F) → Γ(F) such that ∇X − ∇e X (σ) =

A(X, σ). By the compactness of M and the boundedness of A, we conclude |σ| m ∼ |σ| m . Therefore J F,∇F J F,∇e F the topology of W m,pΓ(F) is uniquely defined.

0 ∗ Definition 56 (Distributions). Set DΓ(F) = Γ00 (F) as the space of test sections and D Γ(F) = (DΓ(F)) the space of distributional sections. As usual, in the category of locally convex TVS, DΓ( ) is given ◦ F Schwartz’s topology as the colimit of {Γ(F)K : K ⊂ M compact}, where Γ (F)K := {σ ∈ Γ(F) : supp σ ⊂ K} has the Frechet C∞ topology.

6.1.2 Compatibility with scalar function spaces We aim to show that the global definitions of Sobolev spaces in Subsection 6.1.1 are compatible with the definitions of Sobolev spaces by local coordinates.

Let (ψα,Uα)α be a finite partition of unity, where Uα is open in M and ψα is supported in Uα. Normally n n n in differential geometry, Uα is diffeomorphic to either R+ ∩BR (0, 1) or BR (0, 1). However, it is problematic that the half-ball does not have C∞ boundary, so we use some piecewise-linear functions and mollification to create a bounded C∞ domain.

39 Figure 2: Smoothing the corners

∞ n So WLOG, Uα is diffeomorphic to the closure of a bounded C domain in R , and scalar function spaces are well-defined on Uα (recall Definition 44). Note that supp ψα might intersect with ∂M. For Uα chosen small enough, the bundle F on Uα is diffeomorphic to Uα × F (where F is the typical fiber of F). ◦  α  α α Let eβ be the coordinate sections on supp ψα, and cut off such that supp ψα ⊂ supp eβ ⊂ supp eβ ⊂ β α ∞ α P α α Uα. Let σ ∈ Γ(F) . Then there exist cβ (σ) ∈ Cc (Uα) such that supp cβ (σ) ⊂ supp ψα, ψασ = β cβ (σ)eβ and X α α σ = cβ (σ)eβ α,β P Now, observe that |σ| ∼ |ψασ| and F α F

1 1   2   2 X α α α α X α 2 |ψασ| =  cβ (σ)cβ0 (σ) eβ , eβ0  ∼  cβ (σ)  F F β,β0 β

D α α E P To see this, let x ∈ supp ψα and eβ , eβ0 (x) = Bββ0 (x). Then Bx(u, v) := β,β0 uβvβ0 Bββ0 (x) is a positive-definite inner product, which inducesF a norm on a finite-dimensional vector space, where all norms

are equivalent. Then simply note Bx(u, u) is continuous in variable x ∈ supp ψα. γ ∞ F α P γ α Also, in local coordinates, there are siβ ∈ Cc (Uα) such that ∇i eβ = γ siβeγ on supp ψα. Then

F X α α X α γ α X α α α α X α β ∇i (ψασ) = ∂icβ (σ)eβ + cβ (σ)siβeγ = diβ(σ)eβ where diβ(σ) = ∂icβ (σ) + cγ (σ)siγ β β,γ β γ

P α P α P α P α So |σ|J 1 ∼ α,β |cβ (σ)| + α,β,i |diβ(σ)| ∼ α,β |cβ (σ)| + α,β,i |∂icβ (σ)|. F P P (k) α Similarly |σ| m ∼ ∇ c (σ) . J F α,β k≤m β So for m ∈ N0, p ∈ [1, ∞), X α kσkW m,p ∼ cβ (σ) m,p W (Uα,R) α,β

 α   α P α α Now define Sσ = cβ (σ) and R cβ = α,β cβ eβ . Then RS = 1 on Γ(F) and P := SR is a α,β α,β Q ∞ projection on α,β C (Uα). Note that P depends on the choice of partition of unity. By looking into the Q 1 definitions of R and S, we can extend this to have P = SR as a continuous projection on α,β L (Uα) and

α
X α α m,p P cβ . cβ m,p for m ∈ N0, p ∈ [1, ∞] , cβ ∈ W (Uα) α,β Q W m,p(U ) W (Uα) α,β α α,β

The keen reader should have noticed we never mentioned the case p = ∞ in Subsection 6.1.1 as we defined W m,pΓ(F) by the completion of smooth sections, and C∞(M) is not dense in W m,∞(M). Now,

40 m,p P α α α m,p however, by using local coordinates, we are justified in defining W Γ(F) = { α,β cβ eβ : cβ ∈ W (Uα)} for m ∈ N0, p ∈ [1, ∞] with the norm defined (up to equivalent norms) as

X α α X α α c e := S c e β β β β α,β m,p α,β Q m,p W ΓF α,β W (Uα)

s s Then Bp,qΓ(F) and Fp,qΓ(F) can be defined similarly. In other words, for m ∈ N0, p ∈ [1, ∞], q ∈ [1, ∞], s ≥ 0:

m,p Q m,p • W Γ(F) ' P α,β W (Uα) s Q s • Bp,qΓ(F) ' P α,β Bp,q(Uα) s Q s • Fp,qΓ(F) ' P α,β Fp,q(Uα), p 6= ∞ By using Blackbox 46, we can show the Banach topologies of these spaces are uniquely defined (independent of the choices of ψα,Uα). For convenience (such as working with Holder’s inequality), we still use the Sobolev m,p norms W (m ∈ N0, p ∈ [1, ∞)) defined globally in Subsection 6.1.1. All theorems from section5 that worked on bounded C∞ domains carry over to our setting on M, mutatis 1 3 3 mutandis. For instance, B3,1Γ(F)  L Γ(F)|∂M is a continuous surjection and

1 3 3 1,3
B3,1Γ(F) = L Γ(F),W Γ(F) 1 3 ,1

∗ 0 Moreover, for p ∈ (1, ∞), LpΓ(F) is reflexive. By Holder’s inequality, (LpΓ(F)) = Lp Γ(F) for p ∈ (1, ∞).

6.1.3 Complexification issue

A small step which we omitted is complexification. As F is a real vector bundle, the previous definitions only m,p Q m,p give W Γ(F) ' P α,β W (Uα, R) for m ∈ N0, p ∈ [1, ∞]. In working with real manifolds, differential forms/tensors and their dot products, we always assume real-valued coefficients for sections, but whenever we need to use theorems involving complex Banach spaces or the theory of function spaces, we assume an implicit complexification step. Fortunately, no complications arise from complexification (see SectionA for the full reasoning), so for the rest of the paper we can ignore this detail. When we want to be explicit, we will specify the scalars we are using, e.g. RW m,pΓ(F) versus CW m,pΓ(F).

6.2 Differential forms & boundary Unless mentioned otherwise, the metric is the Riemannian metric, and the connection is the Levi-Civita connection. For X ∈ XM, define nX = hX, νi ν ∈ XM| (the normal part) and tX = X| −nX (the tangential ∂M ∂M part). We note that tX and nX only depend on X ∂M , so t and n can be defined on XM ∂M , and by abuse ∼ of notation, t (XM|∂M ) −→ X(∂M). For ω ∈ Ωk (M) , define tω and nω by

tω(X1, ..., Xk) := ω(tX1, ..., tXk) ∀Xj ∈ XM, j = 1, ..., k

k
∼ k and nω = ω|∂M − tω. By abuse of notation, we similarly observe that t Ω (M) ∂M −→ Ω (∂M) = ∗ k
∗ k
 Ω (M) ∂M =  Ω (M) .

41 [ ] ∗ Recall the musical isomorphism: Xp(Yp) = hXp,Ypi and ωp,Yp = ωp(Yp) for p ∈ M, ωp ∈ Tp M,Xp ∈ TpM,Yp ∈ TpM. Recall the usual Hodge star operator ? :Ωk(M) −→∼ Ωn−k(M), exterior derivative d :Ωk(M) → Ωk+1(M), codifferential δ :Ωk(M) → Ωk−1(M), and Hodge Laplacian ∆ = − (dδ + δd) (cf. [Tay11a, Section 2.10] and [Sch95, Definition 1.2.2]). We will often use Penrose abstract index notation (cf. [Wal84, Section 2.4]), which should not be confused with the similar-looking Einstein notation for local coordinates, or the similar-sounding Pen- rose graphical notation. In Penrose notation, we collect the usual identities in differential geometry (cf. [Lee09]):

• For any tensor T , define (∇T ) = ∇ T and div T = ∇iT . a1...ak ia1...ak i a1...ak ia2...ak • (dω) = (k + 1) ∇ ω ∀ω ∈ Ωk(M) where ∇ is any torsion-free connection. ba1...ak e [b a1...ak] e • (δω) = −∇bω = −(div w) ∀ω ∈ Ωk(M) a1...ak−1 ba1...ak−1 a1...ak−1 ij i σj j iσ σ ij σ ij ij • (∇a∇b − ∇b∇a) T kl = −Rabσ T kl − Rabσ T kl + Rabk T σl + Rabl T kσ for any tensor T kl, where R is the Riemann curvature tensor and ∇ the Levi-Civita connection. Similar identities hold for other types of tensors. When we do not care about the exact indices and how they contract, we ij can just write the schematic identity (∇a∇b − ∇b∇a) T kl = R ∗ T. As R is bounded on compact M, interchanging derivatives is a zeroth-order operation on M.

• For tensor Ta1...ak , define the Weitzenbock curvature operator

k X i X σ X µ σ Ric(T )a1...ak = 2 ∇[i∇aj ]Ta1...aj−1 aj+1...ak = Raj Ta1...aj−1σaj+1...ak − Raj al Ta1...σ...µ...ak j=1 j j6=l

σ where Rab = Raσb is the Ricci tensor. The invariant form is

X i Ric(T )(X1, ...Xk) = (R(∂i,Xa)T )(X1, ..., Xa−1, ∂ ,Xa+1, ..., Xk) ∀Xj ∈ XM a

i ij where ∂ = g ∂j and R(∂i, ∂j) = ∇i∇j −∇j∇i (Penrose notation). Note that hR(∂a, ∂b)∂d, ∂ci = Rabcd. ∞ σ Special cases include Ric(f) = 0 ∀f ∈ C (M) and Ric(X)a = Ra Xσ ∀X ∈ XM (justifying the notation Ric). In local coordinates j i
k Ric (ω) = dx ∧ R(∂i, ∂j)ω · ∂ ∀ω ∈ Ω (M), where · stands for contraction (interior product). Then we have the Weitzenbock formula:

i k ∆ω = ∇i∇ ω − Ric(ω) ∀ω ∈ Ω (M)

i 2 where ∇i∇ ω = tr(∇ ω) is also called the connection Laplacian, which differs from the Hodge Laplacian by a zeroth-order term. The geometry of M and differential forms are more easily handled by the Hodge Laplacian, while the connection Laplacian is more useful in calculations with tensors and the Penrose notation.

a1...ak • For tensors Ta1...ak and Qa1...ak , the tensor inner product is hT,Qi = Ta1...ak Q . But for ω, η ∈ Ωk(M), there is another dot product, called the Hodge inner product, where

1 hω, ηi = hω, ηi Λ k!

42 q 1 R R So |ω|Λ = k! |ω| . Then we define hhω, ηii = M hω, ηi vol and hhω, ηiiΛ = M hω, ηiΛ vol. Recall that k k ω ∧ ?η = hω, ηiΛ vol ∀ω ∈ Ω (M), ∀η ∈ Ω (M). Also

k k+1 hhdω, ηiiΛ = hhω, δηiiΛ ∀ω ∈ Ω00(M), ∀η ∈ Ω00 (M)

So h·, ·iΛ is more convenient for integration by parts and the Hodge star. Nevertheless, as they only m,p k differ up to a constant factor, we can still define W Ω (M)(m ∈ N0, p ∈ [1, ∞)) by h·, ·i as in Subsection 6.1. Finally, by the Weitzenbock formula and Penrose notation, we easily get the Bochner formula: 1 1 ∆ |ω|2
= ∇ ∇i (hω, ωi) = h∆ω, ωi + |∇ω|2 + hRic (ω) , ωi 2 2 i

Λ i W Remark. In [Sch95], the conventions are a bit different, with ∆ = (dδ + δd) , ∆ = −∇i∇ ,R = − Ric and

N the inwards unit normal vector field. Also the difference between hh·, ·ii and hh·, ·iiΛ is not made explicit in the book. We will not use such notation. Lemma 57. Some basic identities:

k ∗ 1. ∀ω ∈ Ω (M): tω = 0 ⇐⇒  ω = 0 . Similarly, nω = 0 ⇐⇒ ιν ω = 0. 2. (tX)[ = t(X[) ∀X ∈ XM

∗ ∗ [ [ k l 3.  tω =  ω, tω = ιν (ν ∧ ω), nω = ν ∧ ιν ω, t(ω ∧ η) = tω ∧ tη ∀ω ∈ Ω (M), ∀η ∈ Ω (M)

k 4. hhtω, ηiiΛ = hhtω, tηiiΛ = hhω, tηiiΛ ∀ω, η ∈ Ω (M) 5. t (?ω) = ? (nω), n (?ω) = ? (tω), ?dω = (−1)k+1 δ ? ω, ?δω = (−1)k d ? ω, ?∆ω = ∆ ? ω ∀ω ∈ Ωk(M) 6. ∗tdω = ∗dω = d∂M ∗ω = d∂M ∗tω ∀ω ∈ Ωk(M) 7. Let ω ∈ Ωk(M). If tω = 0 then tdω = 0. If nω = 0 then nδω = 0.

k 8. ιν ω = t (ιν ω) = ιν nω ∀ω ∈ Ω (M)

∗ ∗ ∗ k k+1 9.  (ω ∧ ?η) = h ω,  ιν ηiΛ vol∂ ∀ω ∈ Ω (M), ∀η ∈ Ω (M)

∗ [ Proof. We will only prove the last assertion. Observe that  (vol) = 0 so vol |∂M = n vol = ν ∧ιν vol. ∗ k ∼ ∗ k Recall vol∂ =  (ιν vol) and tΩ −→  Ω , so the problem is equivalent to proving

[ ν ∧ tω ∧ t ? η = htω, tιν ηiΛ vol on ∂M

Simply observe that tιν η = ιν η and

[ [ D [ E D [ [ E ν ∧ tω ∧ t (?η) = ν ∧ tω ∧ ?nη = ν ∧ tω, nη vol = ν ∧ tω, ν ∧ ιν η vol = htω, ιν ηiΛ vol Λ Λ

Theorem 58 (Integration of tensors and forms by parts).

1. For tensors Ta1...ak and Qa1...ak+1 , Z Z Z Z ia1...ak
ia1...ak ia1...ak ia1...ak ∇i Ta1...ak Q = ∇iTa1...ak Q + Ta1...ak ∇iQ = νiTa1...ak Q M M M ∂M (8) R R R In other words, M h∇T,Qi vol + M hT, div Qi vol = ∂M hν ⊗ T,Qi vol∂ .

43 0 2. For p ∈ (1, ∞), ω ∈ RW 1,pΩk, η ∈ RW 1,p Ωk+1 :

∗ ∗ hhdω, ηiiΛ = hhω, δηiiΛ + hh ω,  ιν ηiiΛ (9)

∗ ∗ R ∗ ∗ where hh ω,  ιν ηiiΛ = ∂M h ω,  ιν ηiΛ vol∂ .

0 3. For p ∈ (1, ∞), ω ∈ RW 2,pΩk(M), η ∈ RW 1,p Ωk(M):

∗ ∗ ∗ ∗ D(ω, η) = hh−∆ω, ηiiΛ + hh ιν dω,  ηiiΛ − hh δω,  ιν ηiiΛ (10)

where D(ω, η) := hhdω, dηiiΛ + hhδω, δηiiΛ is called the Dirichlet integral.

Proof.

i ia1...ak 1. Let X = Ta1...ak Q . Then it is just the divergence theorem. 2. By approximation, it is enough to prove the smooth case. Z Z Z ∗ ∗ ∗ h ω,  ιν ηiΛ vol∂ =  (ω ∧ ?η) = d (ω ∧ ?η) ∂M ∂M M Z Z k = dω ∧ ?η + (−1) ω ∧ d ? η = hhdω, ηiiΛ − hhω, δηiiΛ M M

3. Trivial.

6.3 Boundary conditions and potential theory Definition 59. We define:

k k • ΩD(M) = {ω ∈ Ω (M): tω = 0} (Dirichlet boundary condition)

k k • ΩhomD(M) = {ω ∈ Ω (M): tω = 0, tδω = 0} (relative Dirichlet boundary condition)

k k • ΩN (M) = {ω ∈ Ω (M): nω = 0} (Neumann boundary condition)

k k • Ωhom N (M) = {ω ∈ Ω (M): nω = 0, ndω = 0} (absolute Neumann boundary condition)

k k k • Ω0 (M) = ΩD (M) ∩ ΩN (M)(trace-zero boundary condition) •H k(M) = {ω ∈ Ωk(M): dω = 0, δω = 0} (harmonic fields)

k k k •H D(M) = H (M) ∩ ΩD(M)(Dirichlet fields)

k k k •H N (M) = H (M) ∩ ΩN (M)(Neumann fields) Remark. In writing the function spaces, we omit M when there is no possible confusion. Note that Ωk ◦ 00 k (compact support in M) is different from Ω0 . We can readily extend these definitions to less regular spaces by replacing ω ∈ Ωk with, for example, 1 3 k ω ∈ B3,1Ω . Boundary conditions are defined via the trace theorem, and therefore require some regularity. 1 3 k 2 k 1 k For example, B3,1ΩN makes sense, while L ΩN and H ΩhomN do not make sense. 2 k
2 2 k k 2 k Observe that L -cl ΩN (closure in the L norm) is just L Ω since Ω00 is dense in L Ω .

44 Most of these symbols come from [Sch95]. Note that in [Sch95], the difference between L2X and L2-cl(X) (where X is some space) is not made explicit. Function spaces of type p = ∞ are problematic since the smooth members are not dense (see Corollary 48). For instance, W m,∞Ωk 6= W m,∞-cl(Ωk) in general. 0 0 ∞ 0 0 A special case is when k = 0: ΩN (M) = Ω (M) = C (M) and ΩhomD(M) = ΩD(M). Indeed, 0 0 the conditions for ΩhomD and ΩhomN are what analysts often call “Dirichlet” and “Neumann” boundary conditions respectively. 1 1 In fluid dynamics, the condition for ΩN is also called “impermeable”, while Ω0 is “no-slip”. On the other 1 hand, ΩhomN is often given various names, such as “Navier-type”, “free boundary” or “Hodge” [MM09a; 1 Mon13; BAE16]. The consensus, however, seems to be that ΩhomN should be called the “absolute boundary condition” [Wu91; Hsu02; COQ09; Bau17; Ouy17], which explains our choice of naming. Lemma 60. We have Hodge duality:

k ∼ n−k k ∼ n−k k ∼ n−k • ? :ΩD(M) −→ ΩN (M), ? :Ωhom D(M) −→ Ωhom N (M), ? : HD(M) −→HN (M). k •∇ X (?ω) = ? (∇X ω), |?ω|Λ = |ω|Λ for ω ∈ Ω ,X ∈ XM. m,p k ∼ m,p n−k m,p k ∼ • For m ∈ N0, p ∈ [1, ∞), we have ? : W ΩD(M) −→ W ΩN (M), ? : W Ωhom D(M) −→ m,p n−k W Ωhom N (M). We stress that harmonic fields are harmonic forms, i.e. ∆ω = 0, but the converse is not true in general. Theorem 61 (4 versions). Let ω ∈ Ωk(M) be a harmonic form. Then ω is a harmonic field if either 1. tω = 0, nω = 0 (trace-zero) 2. tω = 0, tδω = 0 (relative Dirichlet) 3. nω = 0, ndω = 0 (absolute Neumann) 4. tδω = 0, ndω = 0 (Gaffney)

Proof. Trivial to show D(ω, ω) = 0 via integration by parts.

Remark. The four conditions correspond to four different versions of the Poisson equation ∆ω = η (cf. [Sch95, Section 3.4]), and four ways we can make ∆ self-adjoint. In this paper, we will just focus on the absolute Neumann Laplacian and the absolute Neumann heat flow. Gaffney, one of the earliest figures in the field, showed that the Laplacian corresponding to the 4th bound- ary condition is self-adjoint and called it the “Neumann problem” (cf. [Gaf54; Con54]). We, however, feel the name “Neumann” should only be used when its Hodge dual is Dirichlet-related (for instance, the Dirichlet potential vs the Neumann potential, to be introduced shortly). Therefore, absent a better rationalization or convention, we see no reason not to honor the name of the mathematician. In the same vein, some authors consider the 1st condition to be the “Dirichlet boundary condition” (following the intuition from the scalar case, where the trace and the tangential part coincide). By the same reasoning as above, we choose not to do so in this paper. k k Blackbox 62 (Dirichlet/Neumann fields). HD(M) and HN (M) are finite-dimensional, and therefore com- m,p k plemented in RW Ω (M) ∀m ∈ N0, p ∈ [1, ∞]. k k Remark. All norms on HN are equivalent, so we do not need to specify which norm on HN we are using at any time. These are very nice spaces, yet they often prevent uniqueness for boundary value problems. We almost always want to work on their orthogonal complements, where Hodge theory truly shines.

45 Proof. See [Sch95, Theorem 2.2.6].

m,p k k Corollary 63. ∀m ∈ N0, p ∈ [1, ∞], there is a continuous projection Pm,p : RW Ω  HN such that

m0,p0 k • it is compatible across different Sobolev spaces, i.e. Pm0,p0 (ω) = Pm1,p1 (ω) if ω ∈ W Ω ∩ W m1,p1 Ωk.

m,p k m,p k
⊥ m,p k k • 1 − Pm,p : RW Ω  W HN := {ω ∈ W Ω : hhω, φiiΛ = 0 ∀φ ∈ HN } is also a compatible projection.

m,p k k
∗ Proof. Define the continuous linear map Im,p : W Ω → HN where

k m,p k Im,pω(φ) = hhω, φiiΛ ∀φ ∈ HN , ∀ω ∈ W Ω

k Then note that (φ1, φ2) 7→ hhφ1, φ2iiΛ is a positive-definite inner product on HN , so Im,p k : HN k ∼ k
∗ HN −→ HN . We also observe that Im,p k does not depend on m, p, so we can define the HN k
∗ ∼ k continuous inverse J : HN −→HN . Then we can just set Pm,p = J ◦ Im,p. As we defined Im,p by hh·, ·iiΛ, Pm,p is compatible across different m, p.

m,p k N N⊥ N k Remark. From now on, for ω ∈ W Ω , we can decompose ω = P ω + P ω where P ω = ω k ∈ H HN N N⊥ m,p k
⊥ and P ω = ω k ⊥ ∈ W HN . The decomposition is natural, i.e. continuous and compatible across (HN ) D D⊥ N⊥ 1,p k 1,p k different Sobolev spaces. By Hodge duality, similarly define P and P . Note P W ΩN ≤ W ΩN N⊥ 2,p k 2,p k and P W Ωhom N ≤ W Ωhom N .

Blackbox 64 (Potential theory). For m ∈ N0, p ∈ (1, ∞), we define the injective Neumann Laplacian

N⊥ m+2,p k N⊥ m,p k ∆N : P W Ωhom N → P W Ω

−1 as simply ∆ under domain restriction. Then (−∆N ) is called the Neumann potential, which is bounded N⊥ m,p k (and actually a Banach isomorphism). ∆N can also be thought of as an unbounded operator on P W Ω . −1 By Hodge duality, we also define the Dirichlet counterparts ∆D and (−∆D) .

Proof. See [Sch95, Section 2.2, 2.3]

Remark. Because duality is involved, we stay away from p ∈ {1, ∞}. Amazingly enough, this is the only elliptic estimate we will need for the rest of the paper. One could say the whole theory is a functional analytic consequence of elliptic regularity (much like how the Nash embedding theorem is a consequence of Schauder estimates, following G¨unther’s approach [Tao16]). There are many identities which might seem complicated, but are actually trivial to check and helpful for grasping the intuition behind routine operations in Hodge theory, as well as its rich algebraic structure.

1,p k 1,p k Definition. We write dc as d restricted to W ΩD and δc as δ restricted to W ΩN for p ∈ (1, ∞). We will prove in Subsection 7.4 that they are essentially adjoints of δ and d. Let us note that ∆N = − (dδc + δcd) N⊥ 2,p k on P W Ωhom N . Corollary 65. Let p ∈ (1, ∞). Some basic properties: 1. PD⊥δ = δ and PN⊥d = d on W 1,pΩk. N⊥ 1,p k D⊥ 1,p k P δc = δc on W ΩN and P dc = dc on W ΩD.

46 −1 −1 D⊥ 1,p k −1 −1 N⊥ 1,p k 2. (−∆D) δ = δ (−∆D) on P W Ω and (−∆N ) d = d (−∆N ) on P W Ω . −1 −1 N⊥ 1,p k −1 −1 D⊥ 1,p k (−∆N ) δc = δc (−∆N ) on P W ΩN and (−∆D) dc = dc (−∆D) on P W ΩD. 3. δ = δPD⊥ = δPN⊥ and d = dPD⊥ = dPN⊥ on W 1,pΩk.

4. dδd = d (δd + dδ) = d (−∆). −1 D⊥ 1,p k −1 N⊥ 1,p k δdδ (−∆D) = δ on P W Ω and dδd(−∆N ) = d on P W Ω .

2,p k
2,p k
1,p k+1 2,p k
2,p k
1,p k−1 5. d W Ωhom N = d W ΩN ∩ W ΩN , δ W Ωhom D = δ W ΩD ∩ W ΩD . 3,p k
2,p k+1 3,p k
2,p k−1 d W Ωhom N ≤ W Ωhom N , δ W Ωhom D ≤ W Ωhom D. −1 Remark. A good mnemonic device is that ∆N is formed by d and δc, so (−∆N ) commutes with d and δc.

Proof.

1. Integration by parts.

2. Just check that the expressions are defined by using 1).

The rest is trivial.

6.4 Hodge decomposition We proceed differently from [Sch95], by using a more algebraic approach in order to derive some results not found in the book. There will be a lot of identities gathered through experience, so their appearances can seem unmotivated at first. Hence, as motivation, let’s look at an example of a problem we will need Hodge 2,p k 1,p k theory for: is it true that W ΩhomN is dense in W ΩN for p ∈ (1, ∞)? The problem is more subtle than it seems, and it is true that the heat flow, once constructed, will imply the answer is yes. But we do not yet have the heat flow, and it turns out this problem is needed for the W 1,p-analyticity of the heat flow itself. This foundational approximation of boundary conditions can be done easily once we understand Hodge theory and the myriad connections between different boundary conditions. m,p k Let ω ∈ W Ω (m ∈ N0, p ∈ (1, ∞)). In one line, the Hodge-Morrey decomposition algorithm is

−1 D⊥ −1 N⊥ ω = dcδ (−∆D) P ω + δcd (−∆N ) P ω + ωHk

D⊥ N⊥ where P ω = ω k ⊥ , P ω = ω k ⊥ are defined as in Corollary 63, and ωHk is simply defined by (HD ) (HN ) subtraction. This is the heart of the matter, and the rest is arguably just bookkeeping. 1,p k −1 N⊥ N⊥ Note that if ω ∈ W Ω , dω = dδd (−∆N ) P ω + dωHk = dP ω + dωHk = dω + dωHk . So dωHk = 0 and similarly δωHk = 0, justifying the notation. A mild warning is that we do not yet have W 1,pHk = W 1,p-cl Hk
. As we will keep referring to this decomposition, let us define

−1 D⊥ −1 D⊥ −1 1,p k •P 1 = dcδ (−∆D) P . Then P1 = dc (−∆D) δP = dc (−∆D) δ on W Ω .

−1 N⊥ −1 1,p k •P 2 = δcd (−∆N ) P Then P2 = δc (−∆N ) d on W Ω .

•P 3 = 1 − P1 − P2.

We observe that the decomposition 1 = P1 + P2 + P3 is natural (continuous and compatible across different Sobolev spaces) since all the operations are natural. In particular, Pj (for j ∈ {1, 2, 3}) is a zeroth-order operator, and if ω is smooth, so is Pjω by Sobolev embedding. Recall that tω = 0 implies tdω = 0, while nω = 0 implies nδω = 0 (Lemma 57).

47 k Theorem 66 (Smooth decomposition). Some basic properties of Pj on Ω :

k+1 k−1 1. P1δ = 0 on Ω and P2d = 0 on Ω . k P1 = P2 = 0 on H .

k+1 k−1 2. P3δc = 0 on ΩN and P3dc = 0 on ΩD . k L3 k
3. PjPi = δijPi. Therefore Ω = j=1 Pj Ω .

k
k−1
D⊥ k−1
D⊥ k
k 4. P1 Ω = dc ΩD = dcP ΩD = dcδP Ωhom D ≤ ΩD. k
k+1
N⊥ k+1
N⊥ k
k P2 Ω = δc ΩN = δcP ΩN = δcdP Ωhom N ≤ ΩN . k
k P3 Ω = H .

k L3 k
5. Ω = j=1 Pj Ω is hh·, ·iiΛ-orthogonal decomposition.

Proof.

k+1 −1 1. On Ω , P1δ = dc (−∆D) δδ = 0. k −1 Let η ∈ H . Then P1η = dc (−∆D) δη = 0.

k+1 −1 N⊥ −1 N⊥ 2. We just need P2δc = δc on ΩN . Indeed, P2δc = δcd (−∆N ) δcP = δcdδc (−∆N ) P = N⊥ δcP = δc.

3. By 1), P2P1 = P1P2 = P1P3 = P2P3 = 0. By 2), P3P2 = P3P1 = 0. Then observe 2 2 2 P2 = (P1 + P2 + P3)P2 = P2 . Similarly, P1 = P1 and P3 = P3.

k
k k
k
4. Recall P3 Ω ≤ H . It becomes an equality since P2 H = P1 H = 0. k
D⊥ k
k−1
Similarly, obviously P1 Ω = dcδP Ωhom D ≤ dc ΩD . It becomes an equality since P2d = 0 and P3dc = 0.

5. Trivial.

To extend this to Sobolev spaces, we will need to use distributions and duality.

m,p k Corollary 67 (Sobolev version). Some basic properties of Pj on W Ω (m ∈ N0, p ∈ (1, ∞)):

m,p k k 1. hhPjω, φiiΛ = hhω, PjφiiΛ ∀ω ∈ W Ω , ∀φ ∈ Ω00, j = 1, 2, 3

m+1,p k+1 m+1,p k−1 2. P1δ = 0 on W Ω and P2d = 0 on W Ω .

m+1,p k m,p k
3. P1 = P2 = 0 on W H and W -cl H .

m+1,p k+1 m+1,p k−1 4. P3δc = 0 on W ΩN and P3dc = 0 on W ΩD . m,p k L3 m,p k
5. PjPi = δijPi. Therefore W Ω = j=1 Pj W Ω .

m,p k
m,p k m,p k
6. P3 W Ω = W H for m ≥ 1 and W -cl H for m ≥ 0. m,p k
m+1,p k+1
N⊥ m+2,p k
P2 W Ω = δc W ΩN = δcdP W Ωhom N . m,p k
m+1,p k−1
D⊥ m+2,p k
P1 W Ω = dc W ΩD = dcδP W Ωhom D .

m+1,p k 7. tP1 = 0 and nP2 = 0 on W Ω .

m,p k L3 m,p k
8. For p ≥ 2,W Ω = j=1 Pj W Ω is hh·, ·iiΛ-orthogonal decomposition.

48 m,p k−1

m+1,p k−1
m,p k+1

m+1,p k+1
9. W -cl dc ΩD = dc W ΩD and W -cl δc ΩN = δc W ΩN . W m+1,p-cl Hk
= W m+1,pHk.

N⊥ N⊥ m+1,p k 10. d = d (P1 + P2 + P3) = dP2 = dP = P d on W Ω . m+2,p k
m+2,p k
m+2,p k
m+2,p k Consequently, ndP2 W Ωhom N = nd W Ωhom N = 0, and P2 W Ωhom N ≤ W Ωhom N . We also have

m+1,p k−1
m+1,p k−1
m+1,p k−1
N⊥ m+1,p k−1
d W Ω = dP2 W Ω = d W ΩN = dP W ΩN

m+1,p k 11. δc = P2δc on W ΩN and

m+1,p k
m+2,p k+1
N⊥ m+3,p k
N⊥ m+2,p k+1
P2 W Ω = δc W ΩN = δcdP W Ωhom N = δcP W Ωhom N

Remark. Note that Lp-cl Hk
(p ∈ (1, ∞)) is defined, while LpHk is not.

Proof.

m+1,p k−1
m+1,p k+1
m,p k
1. Observe P1ω ∈ dc W ΩD , P2ω ∈ δc W ΩN , P3ω ∈ W -cl H . Simply m+1,p k−1
k+1
m,p k
k−1
show dc W ΩD ⊥ δc ΩN , W -cl H ⊥ dc ΩD , and so forth via integration by parts.

2. W m+1,pΩk+1 = W m+1,p-cl Ωk+1
.

m,p k
m+1,p k k 3. The case W -cl H is trivial. For ω ∈ W H , hhP1ω, φiiΛ = hhω, P1φiiΛ = 0 ∀φ ∈ Ω00 m+1,p k k−1 since W H ⊥ dc(ΩD ) (integration by parts).

m+1,p k+1 k m+1,p k+1
4. Let ω ∈ W ΩN . Then hhP3δcω, φiiΛ = hhδcω, P3φiiΛ = 0 ∀φ ∈ Ω00 since δc W ΩN ⊥ Hk.

The rest is trivial.

To connect Hodge decomposition to fluid dynamics, we will need the Friedrichs decomposition:

N N⊥
N ex P3 = P + P P3 = P3 + P3

where

N N N N N⊥ N⊥ •P 3 := P P3 = P = P3P (as P P1 = P1 and P P2 = P2)

ex N⊥ N⊥ •P 3 := P P3 = P3P

D co We similarly define P3 , P3 via Hodge duality. Note that ex and co stand for “exact” and “coexact” (and we will see why shortly). N ex N
Then we define P := P3 + P2 as the Leray projection. Then 1 = (P3 + P1) + P3 + P2 = ex (P3 + P1) + P is called the Helmholtz decomposition.

N ex m,p k Theorem 68 (Friedrichs decomposition). Basic properties of P3 , P3 on W Ω (m ∈ N0, p ∈ (1, ∞)):

ex −1 ex m,p k 1. P3 = dδ(−∆N ) P3 on W Ω .

ex m,p k
m,p k
m+1,p k−1
2. P3 W Ω = W -cl H ∩ d W Ω .

ex m,p k
m+1,p k−1
m+1,p k−1
N⊥ m+1,p k−1
3. (P3 + P1) W Ω = d W Ω = d W ΩN = dP W ΩN .

49 m,p k
N
m,p k
  m,p    4. P W Ω = P3 + P2 W Ω = Ker δc m,q k when m ≥ 1 and W -cl Ker δc k W ΩN ΩN when m ≥ 0.

m,p k

m,p

5. (P3 + P2) W Ω = Ker δ W m,q Ωk when m ≥ 1 and W -cl Ker δ Ωk when m ≥ 0. N⊥ N⊥ m,p k 6. P P = P2 = PP on W Ω . N⊥ N⊥ N⊥ m+1,p k Therefore dP = dP P = dP2 = d = dP = P d on W Ω .

m+2,p k
m+2,p k
k m+2,p k 7. P W Ωhom N ≤ P2 W Ωhom N ⊕ HN ≤ W Ωhom N .

Proof.

k −1 ex −1 ex ex −1 ex −1 ex 1. On Ω : δd(−∆N ) P3 = δ(−∆N ) dP3 = 0, so P3 = (−∆)(−∆N ) P3 = dδ(−∆N ) P3 . Then we are done by density.

N N N⊥ 2. P3 d = P3P d = 0 as P d = d.

N 3. P2d = 0 and P3 d = 0.   4. We first prove the smooth version. Let ω ∈ Ker δc k . Then hhP1ω, P1ωiiΛ = hhP1ω, ωiiΛ = ΩN   k−1
ex N
k 0 as Ker δc k ⊥ d Ω , so P1ω = 0. Similarly, P3 ω = 0. Then P3 + P2 Ω = ΩN   Ker δc k . ΩN m,p k m,p    For W Ω , the case W -cl Ker δc k is trivial. Then assume m ≥ 1 and ω ∈ ΩN   ex   Ker δc m,q k . We can show P1ω = P3 ω = 0 as distributions since Ker δc m,q k ⊥ W ΩN W ΩN d(Ωk−1).

k−1 5. Just note that Ker δ W m,q Ωk ⊥ dc(ΩD ) and argue similarly. N⊥ N N N⊥ N⊥ N⊥ 6. Easy to check that P P3 = P3 P = 0 and P P2 = P2P = P2. 7. Trivial.

D co k k k Remark. Similar results for P3 , P3 hold by Hodge duality. When M has no boundary, H = HD = HN so N D P3 = P3 = P3 . A simple consequence of the Hodge-Helmholtz decomposition is that  

Ker δc k N
k
k

Ω P + P2 Ω (P3 + P1) Ω Ker d k N = 3 = PN Ωk
= = Ω k+1
P (Ωk) 3 (Pex + P ) (Ωk) d (Ωk−1) δc ΩN 2 3 1

k k k k This can be rewritten as Ha (M) = HN (M) = HdR (M, d) (Hodge isomorphism theorem) where HdR (M, d) := !

  Ker δc Ker d Ωk Ωk k N d(Ωk−1) is called the k-th de Rham cohomology group, and Ha (M) := k+1 is called the k-th δc(ΩN ) k k k absolute de Rham cohomology group. In particular, β (M) := dim HN (M) = dim HdR (M, d) is called !

Ker dc Ωk n−k k D the k-th Betti number of M. Note that the Hodge dual of Ha (M) is Hr (M) := k−1 , the k-th dc(ΩD ) relative de Rham cohomology group.

50 We can also define right inverses (potentials) for d, δ, δc, dc (see Subsection 8.1). In many ways, Hodge theory reduces otherwise complicated boundary value problems into purely alge- braic calculations. A standard Hodge-theoretic calculation related to the Euler equation is given later in Subsection 8.2. We can also derive a general form of the Poincare inequality:

N⊥ m+1,p k Corollary 69 (Poincare-Hodge-Dirac inequality). Let ω ∈ P W ΩN (m ∈ N0, p ∈ (1, ∞)). Then

kωkW m+1,p ∼ kdωkW m,p + kδcωkW m,p and we have a bijection

N⊥ m+1,p k d⊕δc m+1,p k
m+1,p k
ex m,p k+1 m,p k−1 P W ΩN −−−→ d W Ω ⊕ δc W ΩN = (P1 + P3 )(W Ω ) ⊕ P2(W Ω )

−1 N⊥ m+1,p k In particular, (d ⊕ δc) (dη, δcυ) = P2 (η − υ) + υ ∀η, υ ∈ P W ΩN .

Proof. Observe that

N⊥ m+1,p k d⊕δc N⊥ m+1,p k
N⊥ m+1,p k
•P W ΩN −−−→ dP W ΩN ⊕ δcP W ΩN is a continuous injection.

N⊥ m+1,p k
m+1,p k
ex m,p k+1 • dP W ΩN = d W Ω = (P1 + P3 )(W Ω ) by Corollary 67.

N⊥ m+1,p k
m+1,p k
m,p k−1 • δcP W ΩN = δc W ΩN = P2(W Ω ) by Corollary 65 and 67.

By open mapping, we only need to prove d⊕δc (the injective Hodge-Dirac operator) is surjective: N⊥ m+1,p k N⊥ m+1,p k let η, υ ∈ P W ΩN . We want to find ω ∈ P W ΩN such that dω = dη, δcω = δcυ. By the restriction δcω = δcυ, the freedom is in choosing

N⊥   N⊥ m+1,p k m+1,p k ϑ := ω − υ ∈ P Ker δc m+1,p k = P P(W Ω ) = P2(W Ω ) W ΩN

such that dω = dυ + dϑ = dη. In other words, we want ϑ such that dϑ = d (η − υ) and P2ϑ = ϑ. Then we are done by setting ϑ = P2 (η − υ).

Remark. We note that a less general version of the Poincare inequality was used in [Sch95] to establish the potential estimates in Blackbox 64 as well as Blackbox 62. A more general version [Sch95, Lemma 2.4.10] deals with the case p ≥ 2. Our version here only requires p ∈ (1, ∞). Among other things, the inequality allows the following approximation of boundary conditions, which will play a crucial role for the W 1,p-analyticity of the heat flow in Subsection 7.3. Corollary 70. Let p ∈ (1, ∞) .

1,p k 2,p k−1
  k k−1
  1. W ΩN = d W ΩhomN ⊕ Ker δc 1,p k and ΩN = d ΩhomN ⊕ Ker δc k . W ΩN ΩN p k

1,p k
1,p k
2. L -cl d ΩhomN = d W ΩN = d W Ω . 1,p 2,p k
1,p k 3. W -cl W ΩhomN = W ΩN .

Proof.

1,p k 1,p k 1,p k 1,p k ex 1,p k 1. Because PW Ω ≤ W ΩN , we conclude PW Ω = PW ΩN . Meanwhile, (P1 + P3 ) W ΩN = 1,p k 1,p k ex 1,p k 2,p k−1
1,p k 2,p k−1
(1 − P) W ΩN ≤ W ΩN , so (P1 + P3 ) W ΩN ≤ d W ΩN ∩W ΩN = d W ΩhomN .

51 p ex k+1
ex p k+1
ex p k+1 2. L -cl (P1 + P3 )ΩN = (P1 + P3 ) L -cl ΩN = (P1 + P3 ) L Ω .

1,p N⊥ k
N⊥ 1,p k 3. We are done if W -cl P ΩhomN = P W ΩN . k
k k
N⊥ k
Recall P2 ΩhomN ≤ ΩhomN and δc ΩN = δcP Ωhom N by Corollary 67, so by the −1 formula of (d ⊕ δc) from Corollary 69:

−1  k
k
 −1  N⊥ k
N⊥ k
 N⊥ k (d ⊕ δc) d ΩhomN ⊕ δc ΩN = (d ⊕ δc) dP ΩhomN ⊕ δcP Ωhom N = P ΩhomN

So

1,p N⊥ k
−1  p k

p k

 W -cl P ΩhomN = (d ⊕ δc) L -cl d ΩhomN ⊕ L -cl δc ΩN −1  1,p k
1,p k
 = (d ⊕ δc) d W Ω ⊕ δc W ΩN N⊥ 1,p k = P W ΩN

6.5 An easy mistake

k Let p ∈ (1, ∞) , ω ∈ ΩhomN . In other words, nω = 0 and ndω = 0. Using intuition from Euclidean space, it is tempting to conclude ∇ν ω = 0, but this is not true in general. We will not use Penrose notation but work in local coordinates on ∂M, with ∂1, ...∂n−1 for directions on ∂M and ∂n for the direction of νe. Let {a1, ..., ak} ⊂ {1, ..., n − 1}. Observe that ndω = 0 implies X 0 = (dω) = ∂nωa ...a + (±1)∂a ω = ∂nωa ...a na1...ak 1 k i na1...abi...ak 1 k i

since ω = 0 on ∂M. Then recall ∂nωa ...a = (∇nω) + Γ ∗ ω where Γ ∗ ω is schematic for some na1...abi...ak 1 k a1...ak terms with the Christoffel symbols. As Γ is bounded on M, we conclude |t∇ν ω| . |ω| and |t∇ν ω|Λ . |ω|Λ on ∂M. Then 2
ιν d |ω| = ∇ν hω, ωi = 2 h∇ν ω, ωi = 2 ht∇ν ω, ωi

2
2 p so ∇ν |ω| . |ω| on ∂M. This will be important in establishing the L -analyticity of the heat flow in Subsection 7.2.

7 Heat flow

As promised, we now obtain a simple construction of the heat flow. We still work on the same setting as in Subsection 6.1.

7.1 L2-analyticity

N⊥ 2 k −1 Recall that ∆N is an unbounded operator on RP L Ω and (−∆N ) is bounded. It is trivial to check −1 that (−∆N ) is symmetric, therefore self-adjoint. Then ∆N is also self-adjoint. Then for ω ∈ D(∆N ) = N⊥ 2 k P H ΩhomN : hh∆N ω, ωiiΛ = −D(ω, ω) ≤ 0. So ∆N is dissipative. Therefore, by a complexification  C  C t∆N argument , ∆N is acutely sectorial of angle 0 by Theorem 41 and e is a C0, analytic semigroup on t≥0 CPN⊥L2Ωk. By Blackbox 38, we can derive some basic facts about et∆N : m N⊥ 2m k m m • For m ∈ 1,D(∆N ) ≤ P H Ω and k∆N ωk 2 ∼ kωk 2m ∼ kωk m ∀ω ∈ D (∆N ) by poten- N L H D(∆N ) t∆N
m tial estimates. Recall that e t≥0 on (D(∆N ), k·kH2m ) is also a C0 semigroup by Sobolev tower

52 (Theorem 32).

• For t > 0, by either the spectral theorem (with a complexification step) or semigroup theory, et∆N N⊥ 2 k ∞ N⊥ k is a self-adjoint contraction on RP L Ω , with image in D(∆N ) ≤ P Ω by the analyticity of  s∆C  e N . s≥0

N⊥ 2 k N⊥ k t∆N
∞ •∀ ω ∈ P L Ω , (0, ∞) → P Ω , t 7→ e ω is C -continuous by Sobolev tower. Let m ∈ N1, m m t∆N
m t∆N t∆N m t∆N m then ∂t e ω = ∆N e ω and e ω H2m ∼ ∆N e ω L2 .¬m,¬t tm kωkL2

2 k Next we define the non-injective Neumann Laplacian ∆gN as an unbounded operator on L Ω with m m m k m D(∆gN ) = D(∆N ) ⊕ HN and ∆gN = ∆N ⊕ 0 ∀m ∈ N1. By using either the spectral theorem or checking the definitions manually, ∆gN is also a self-adjoint, dissipative operator. Then we also get an analytic heat t∆gN t∆N flow, and ∆N = ∆N ⊕ 0 k with e = e ⊕ Id k . g HN HN m,p k N⊥ N Recall that for m ∈ N0, p ∈ (1, ∞) , ω ∈ W Ω : kωkW m,p ∼ P ω m,p + P ω k where we do W HN k not need to specify the norm on HN as they’re all equivalent. Then the previous results for ∆N can easily be extended to ∆gN : m m m 2m k N⊥ m • For m ∈ N1,D(∆gN ) ≤ H Ω and ∀ω ∈ D(∆gN ): ∆gN ω ∼ P ω 2m and kωk ∼ L2 H D(∆gN )   m t∆gN kωkH2m . Recall e on D(∆gN ) is also an a C0 semigroup. (Sobolev tower) t≥0

• For t > 0, by either the spectral theorem (with a complexification step) or semigroup theory, et∆gN is ∞ 2 k   k a self-adjoint contraction on RL Ω , with image in D ∆gN ≤ Ω .   2 k k t∆gN ∞ •∀ ω ∈ L Ω , (0, ∞) → Ω , t 7→ e ω is C -continuous by Sobolev tower. Let m ∈ N1, then   m m t∆gN t∆gN ∂t e ω = ∆gN e ω and

mm t∆gN t∆gN N⊥ N N⊥ N e ω ∼ e P ω + P ω Hk .¬m,¬t m P ω L2 + P ω Hk H2m H2m N t N

t→∞ By these estimates, we conclude that et∆gN −−−→PN in L L2Ωk
(Kodaira projection). In fact, this is how Hodge decomposition was done historically.

7.2 Lp-analyticity

p Though we could use the same symbols ∆N and ∆gN for the Neumann Laplacian on L , that can create confusion regarding the domains. Let them still refer to the unbounded operators on RPN⊥L2Ωk and RL2Ωk as before. However, et∆N and et∆gN are compatible across all Lp spaces (as we will see). ∞ ∞ k     p First we note that Ω00 ≤ D ∆gN so D ∆gN is dense in L ∀p ∈ (1, ∞). Then for Lp-analyticity, we make a Gronwall-type argument (adapted from [IO15, Appendix A] to handle the boundary).  ∞ Theorem 71 (Local boundedness). For p ∈ (1, ∞) , s ∈ (0, 1) and u ∈ D ∆gN :

s∆gN e u .p kukp p

53 ∞   2 p Proof. By duality and the density of D ∆gN in L ∩ L , WLOG assume p ≥ 2. By complex interpolation (with a complexification step), WLOG assume p = 4K where K is a large natural number. s∆N Let U(s) = e u, so ∂sU = ∆U and

4K
4K−2 Bochner 4K−2  2
2  ∂s |U| = 2K|U| h2∆U, Ui = 2K|U| ∆ |U| − 2 |∇U| − 2 hRic (U) ,Ui

So Z Z Z  4K 4K−2 2
4K ∂s |U| ≤ 2K |U| ∆ |U| + OM,K |U| M M M ∞ 2   k Let f = |U| . As U ∈ D ∆gN ≤ Ωhom N , |∇ν f| . f on ∂M by Subsection 6.5. By Gronwall, R 2K−1 R 2K we just need M f ∆f . M f (pseudo-dissipativity). Simply integrate by parts:

2K−1 2K−1
2K−1 ∆f, f = − df, d f + ∇ν f, f Z Z  2 2K−2 2K = −(2K − 1) |df| f + OM f M ∂M Z Z  2K − 1 K
2 2K = − 2 d f + OM f K M ∂M

K R 2 R 2 R 2 Let F = |f| . So for any ε > 0, we want Cε > 0 such that ∂M F ≤ ε M |dF | + Cε M F . This follows from Ehrling’s inequality, and the fact that H1(M) → L2(∂M) is compact.

  So et∆gN can be uniquely extended by density to L2Ωk + LpΩk and et∆gN ∈ L LpΩk
. With t≥0 LpΩk a complexification step and an appropriate core chosen by Sobolev embedding, local boundedness on Lp implies Lp-analyticity for all p ∈ (1, ∞) by Theorem 35.    ∞ t∆gN p k Let Ap be the generator of e on L Ω . By the definition of generator, Ap = ∆gN on D ∆gN . t≥0 C In our terminology, Ap is acutely quasi-sectorial. But we want a more concrete description of D(Ap). ∞   2,p k
2,p    Lemma 72. Let p ∈ (1, ∞). Then D(A ), k·k ∼ W Ω , k·k 2,p and W -cl D ∆ = p D(Ap) hom N W gN 2,p k W Ωhom N .

∞   N⊥ ∞ Proof. Observe that ∀u ∈ D ∆gN : P u ∈ D(∆N ) and

N N⊥ N⊥ N N⊥ kukD(A ) = kukp+ ∆gN u ∼ P u k + P u + ∆N P u ∼ P u k + P u 2,p ∼ kukW 2,p p p HN p p HN W

 ∞   Then k·k ∼ k·k 2,p since D ∆ is a dense core in D(A ), k·k (see Lemma 34). D(Ap) W gN p D(Ap) ∞ 2,p    2,p ∞ k This also implies D(Ap) = W -cl D ∆gN = W -cl (D (∆N )) ⊕ HN .

∆N ∞ N⊥ 2,p k
∼ N⊥ p k
p ∞ Recall that D (∆N ) ≤ P W Ωhom N , k·kW 2,p −→ P L Ω , k·kLp . Since L -cl (∆N D (∆N )) = p ∞ N⊥ p k 2,p ∞ −1 N⊥ p k
N⊥ 2,p k L -cl (D (∆N )) = P L Ω , we conclude W -cl (D (∆N )) = (−∆N ) P L Ω = P W Ωhom N and we are done.

p k s∆gN 1 s∆gN √1 So for p ∈ (1, ∞) , s ∈ (0, 1) and u ∈ L Ω : e u . s kukp. That implies e u . kukp W 2,p W 1,p s  p 2,p  0 2  1 by complex interpolation (with complexification), using CL , CW 1 = CFp,2, CFp,2 1 = CFp,2. 2 2

54 ∞ ∞
k m 2,p k   Obviously, D Ap = {ω ∈ ΩhomN : ∆ ω ∈ W Ωhom N ∀m ∈ N0} = D ∆gN by Sobolev embedding. ∞
p Additionally, by the density of D Ap in L , we can show by approximation that

DD EE DD EE 0 et∆gN ω, η = ω, et∆gN η ∀ω ∈ LpΩk, η ∈ Lp Ωk, p ∈ (1, ∞) , t ≥ 0 Λ Λ

t∆gN N⊥ N⊥ t∆gN m,p k This implies that e P = P e on W Ω ∀m ∈ N0, ∀p ∈ (1, ∞).

7.3 W 1,p-analyticity ∞ ∞ 1,p    1,p  2,p    1,p 2,p k
We first observe that W -cl D ∆gN = W -cl W -cl D ∆gN = W -cl W Ωhom N = 1,p k W ΩN by Corollary 70 and Lemma 72. Ln k Because we will soon be dealing with differential forms of different degrees, define Ω(M) = k=0 Ω (M) as the graded algebra of differential forms where multiplication is the wedge product. We simply define m,p Ln m,p k s s W Ω(M) = k=0 W Ω (M), and similarly for Bp,q,Fp,q spaces. Spaces like ΩD (M), Ω00 (M) or m,p W ΩhomN are also defined by direct sums. The dot products h·, ·iΛ and hh·, ·iiΛ are also definable as the Ln k sum from each degree. Also define H(M) = k=0 H (M). As an example, ω ∈ L2Ω(M) and η ∈ L2Ω(M) would imply ω∧η ∈ L1Ω(M). We also recover integration by parts:

∗ ∗ 1,p 1,p0 hhdω, ηiiΛ = hhω, δηiiΛ + hh ω,  ιν ηiiΛ ∀ω ∈ RW Ω(M) , ∀η ∈ RW Ω(M) , p ∈ (1, ∞)

  2 2,p Then we can set D ∆gN = H Ωhom N and D (Ap) = W Ωhom N (p ∈ (1, ∞)), and previous results such as sectoriality or the Poincare inequality still hold true in this new degree-independent framework, mutatis mutandis.

Theorem 73 (Commuting with derivatives I). Let p ∈ (1, ∞) .

∞

∞
∞

∞
1. δc D Ap ≤ D Ap and d D Ap ≤ D Ap

2,p t∆gN t∆gN 2. Let ω ∈ D(Ap) = W Ωhom N and D ∈ {d, δc, δcd, dδc}. Then for t > 0 : De ω = e Dω.

Proof.

∞
2,p m 2,p 1. Let η ∈ D Ap . Obviously dη ∈ W Ωhom N , so d∆ η ∈ W Ωhom N ∀m ∈ N0. Observe that nη = 0 implies nδη = 0, and ndη = 0 implies nδdη = 0. But n∆η = 0 so 2,p m 2,p ndδη = 0 and we conclude δcη ∈ W Ωhom N . Similarly, δc∆ η ∈ W Ωhom N ∀m ∈ N0.

t∆gN ∞
2. Let t > 0. Note that De ω ∈ D Ap . ∞ h∆gN C   e −1 t∆gN t∆gN t∆gN t∆gN t∆gN Then e ω −−→ ∆gN e ω so ∂t De ω = D∆gN e ω = ∆gN De ω. Therefore h h↓0

eh∆gN Det∆gN ω = De(t+h)∆gN ω ∀t > 0, ∀h > 0

Lp C∞ Note that De(t+h)∆gN ω −−→ Deh∆gN ω since e(t+h)∆gN ω −−→ eh∆gN ω. t↓0 t↓0 Lp W 2,p h∆gN t∆gN h∆gN t∆gN On the other hand, e De ω −−→ e Dω as e ω −−−→ ω (why we need ω ∈ D(Ap)). t↓0 t↓0 So Deh∆gN ω = eh∆gN Dω ∀h > 0.

55 2,p t∆gC t∆gC We can extend this via complexification. For ω ∈ CW ΩhomN , DCe N ω = e N DCω ∀t > 0. p  z∆gC  By L -analyticity, ∃α = α(p) > 0 such that e N is a C , locally bounded, analytic semigroup + 0 z∈Σα ∪{0} C C p C z∆gN z∆gN C + on CL Ω. Then by the identity theorem, D e ω = e D ω ∀z ∈ Σα .

1,p  z∆gC  1,p Theorem 74 (W -analyticity). e N is a C , analytic semigroup on W Ω . + 0 C N z∈Σα ∪{0}

C
1,p
Proof. Note that D(Ap ), k·kW 2,p is dense in CW ΩN , k·kW 1,p by Corollary 70.  z∆gC  1,p
So by Lemma 33, we just need to show e N ⊂ L W Ω and is locally bounded. + C N z∈Σα ∪{0} So it is enough to show

z∆gC + N C
e u . kukW 1,p ∀u ∈ D Ap , ∀z ∈ D ∩ Σα W 1,p

C N⊥ z∆gN N⊥ C
+ Consider P u, then we only need e u . kukW 1,p ∀u ∈ P D Ap , ∀z ∈ D ∩ Σα . W 1,p Recall et∆gN PN⊥ = PN⊥et∆gN from Subsection 7.2. By the Poincare inequality (Corollary 69):

C C C C C z∆gN C z∆gN C z∆gN z∆gN C z∆gN C e u ∼ d e u + δc e u = e d u + e δc u W 1,p p p p p C C N⊥ C
+ . d u p + δc u p ∼ kukW 1,p ∀u ∈ P D Ap , ∀z ∈ D ∩ Σα

1,p t∆gN t∆gN Corollary 75. Let ω ∈ W ΩN and D ∈ {d, δc}. Then for t > 0 : De ω = e Dω.

W 1,p Proof. Same as before, but with et∆gN ω −−−→ ω. t↓0

  t∆gN 1,p 2
Let A1,p be the generator of e on W ΩN . Then A1,p and Ap agree on D Ap by the definition t≥0  ∞  ∞ of generators, so A = ∆ on D ∆ . By potential estimates, k·k ∼ k·k 3,p on D ∆ 1,p gN gN D(A1,p) W gN   ∞ and therefore on k·kW 3,p -cl D ∆gN = D (A1,p). By the same argument as in Lemma 72, D (A1,p) = ∞ −1 N⊥ 1,p
  (−∆N ) P W ΩN ⊕ HN ≥ D ∆gN .

Theorem 76 (Compatibility with Hodge-Helmholtz). Let m ∈ N0, p ∈ (1, ∞), t > 0. By Corollary 75 and Corollary 67:   t∆gN m+1,p
t∆gN m+1,p • e d W ΩN = d e W ΩN ≤ d (ΩN ) = d (Ω).   t∆gN m+1,p
t∆gN m+1,p • e δc W ΩN = δc e W ΩN ≤ δc (ΩN ) = δc (ΩhomN ).

t∆gN t∆gN ex ex t∆gN t∆gN t∆gN As e = 1 on HN , we finally conclude e (P3 + P1) = (P3 + P1) e , e P2 = P2e and t∆gN N N t∆gN N m,p t∆gN t∆gN m,p e P3 = P3 e = P3 on W Ω(M). Also, e P = Pe on W Ω(M) where P is the Leray projection.

56 By the definition of generators,

ex ex N N ∆gN (P3 + P1) = (P3 + P1) ∆gN , P3 ∆gN = ∆gN P3 = 0, P2∆gN = ∆gN P2 = ∆gN P = P∆gN

2,p on D(Ap) = W ΩhomN .

t∆ t∆ We briefly note that in the no-boundary case, we have Ω = ΩN = ΩhomN , ∆gN = ∆gD = ∆, e P1 = P1e m,p 2,p on W Ω, P1∆ = ∆P1 on W Ω. p Remark. The operator P∆gN , with the domain PD(Ap), is a well-defined unbounded operator on PL Ω. By t ∆gN t∆gN p our arguments, its complexification is acutely sectorial, and P∆gN = ∆gN , e P = e on PL Ω. Other authors call it the Stokes operator corresponding to the “Navier-type” / “free” boundary condition [Miy80; Gig82; MM09a; MM09b; BAE16].

7.4 Distributions and adjoints

◦ Like the Littlewood-Paley projection, the heat flow does not preserve compact supports in M. So applying the heat flow to a distribution is not well-defined. This can be a problem as we will need to heat up the nonlinear term in the Euler equation for Onsager’s conjecture. For the Littlewood-Paley projection, we fixed it by introducing tempered distributions. That in turn motivates the following definition.

Definition 77. Let I ⊂ R be an open interval. Define

◦ k k k ∞
• DΩ = Ω00 = colim{ Ω00 (K) ,C topo : K ⊂ M compact} as the space of test k-forms with Schwartz’s topology (colimit in the category of locally convex TVS).

0 k k
∗ • D Ω = DΩ as the space of k-currents (or distributional k-forms), equipped with the weak* topology.

∞ k   ∞ 0 k • DN Ω = D ∆gN as the space of heated k-forms with the Frechet C topology and DN Ω = k
∗ DN Ω as the space of heatable k-currents (or heatable distributional k-forms) with the weak* topology.

k
∞ k
∞ k
∞
• Spacetime test forms: D I, Ω = Cc I, Ω00 = colim{ Cc I1, Ω00(K) ,C topo : I1 × K ⊂ ◦ k
∞ k
∞
I × M compact} and DN I, Ω = colim{ Cc I1, DN Ω ,C topo : I1 ⊂ I compact}.

0 k
k
∗ 0 k
k
∗ • Spacetime distributions D I, Ω = D I, Ω , DN I, Ω = DN I, Ω .

∗ k i k 0 k i 0 k ∗ Obviously DΩ ,−→ DN Ω , so there is an adjoint DN Ω −→ D Ω . Unfortunately, Im(i) is not dense so i ∗ 0 0 is not injective. Nevertheless, we will make i the implicit canonical map from DN to D . In particular, 0 0 DN D k
k
0 k
0 k
ωj −−→ 0 implies ωj −−→ 0. Similarly, D I, Ω ,→ DN I, Ω and DN I, Ω → D I, Ω . C∞ t∆gN k By Sobolev tower (Theorem 32), we observe that e φ −−→ φ ∀φ ∈ D N Ω . t↓0   0 k k t∆gN t∆gN For Λ ∈ DN Ω , t ≥ 0 and φ ∈ D N Ω , we define e Λ(φ) = Λ e φ . As Λ is continuous, ∃m0, m1 ∈

k t∆gN t∆gN N0 such that |Λ(φ)| . kφkCm0 . kφkHm1 . Then for t > 0 and φ ∈ D N Ω : e Λ(φ) . e φ .t,m1 Hm1 t t t∆gN 2 k t∆gN 2 ∆gN 2 ∆gN k kφkL2 =⇒ e Λ ∈ L Ω and e Λ = e e Λ ∈ D N Ω . p k t∆gN p k 0 k Also, for p ∈ (1, ∞) and ω ∈ L Ω , e ω is the same in L Ω and DN Ω . Remark. We note an important limitation: though heated forms are closed under d and δ by Theorem 73, because of integration by parts, we cannot naively define δ or ∆ on heatable currents.

57 0 Analogous concepts such as DD and DD can be defined via Hodge duality for the relative Dirichlet heat flow. Ln k Recall the graded algebra Ω(M) = k=0 Ω (M) from Subsection 7.3. We can easily define DΩ, DN Ω etc. by direct sums. D0 0 0 N DN For Λ ∈ DN Ω and φ ∈ D N Ω, we can define δc Λ(φ) = Λ (dφ) and d Λ(φ) = Λ (δcφ). These will be consistent with the smooth versions, though we take care to note that

DD 0 EE DN ∗ ∗ 1,p δc ω, φ = hhω, dφiiΛ = hhδω, φiiΛ + hh ιν ω,  φiiΛ ∀ω ∈ W Ω, φ ∈ DN Ω, p ∈ (1, ∞) (11) Λ

0 0 D  0 0 0 0  DN 1,p N D DN DN D So δc agrees with δc on W ΩN as defined previously. In particular, ∆gN = − d N δc + δc d N is 0 well-defined on DN Ω. D0 Note that δ N Λ cannot be defined since there is φ ∈ D N Ω such that dcφ is not defined. ε ε∆gN For convenience, we also write Λ (φ) = hhΛ, φiiΛ (abuse of notation) and Λ = e Λ for ε > 0. Observe 0 that for all Λ ∈ DN Ω, φ ∈ DN Ω:

ε ε DD 0  EE ε ε ε DN hhd (Λ ) , φiiΛ = hhΛ , δcφiiΛ = hhΛ, (δcφ) iiΛ = hhΛ, δc (φ )iiΛ = d Λ , φ Λ

ε ε  0   D0  ε DN ε N 0 Then d (Λ ) = d Λ and similarly δc (Λ ) = δc Λ ∀Λ ∈ DN Ω.

p 0 p 0 Problem (Consistency problem). For p ∈ (1, ∞), we have L Ω ,→ DN Ω and L Ω ,→ D Ω, and we can 0 0 0 p 0 p p D DN 0 0 identify DN Ω ∩ L Ω = D Ω ∩ L Ω = L Ω. Let d and d be d defined on D and DN respectively. For 0 0 p D 0 p DN 0 p ω ∈ L Ω, if d ω ∈ D Ω ∩ L Ω , the question is whether we can say d ω ∈ DN Ω ∩ L Ω. p More explicitly, if α, ω ∈ L Ω and hhα, φ0iiΛ = hhω, δcφ0iiΛ ∀φ0 ∈ DΩ, can we say hhα, φiiΛ = hhω, δcφiiΛ ∀φ ∈ DN Ω? The answer is yes, and the method is analogous to some key steps in Subsec- tion 3.3 and Subsection 8.3.

Recall the cutoffs ψr from Equation (3).

p p 1,p k L L Lemma 78. Let p ∈ (1, ∞) and φ ∈ W ΩN . Then (1 − ψr) φ −−→ φ and δc ((1 − ψr) φ) −−→ δcφ. r↓0 r↓0

Proof. In Penrose notation,

δ ((1 − ψ ) φ) = −∇i ((1 − ψ ) φ) = ∇iψ φ − (1 − ψ ) ∇iφ c r a1...ak−1 r ia1...ak−1 r ia1...ak−1 r ia1...ak−1 =⇒ δ ((1 − ψ ) φ) = ι φ + (1 − ψ ) δ φ = f ι φ + (1 − ψ ) δ φ c r ∇ψr r c r νe r c

p L 1 Then we only need frιν φ −−→ 0. As ιν φ = 0 on ∂M, by Theorem 53, kfrιν φkLp kιν φkLp(M ) e r↓0 e e . r e

p 0 p p 0 p Then we can conclude {ω ∈ L Ω(M): dDN ω ∈ L } = {ω ∈ L Ω(M): dD ω ∈ L }. Recall that for an unbounded operator A, we write (A, D(A)) to specify its domain.

Theorem 79 (Adjoints of d, δ). For p ∈ (1, ∞) , the closure of (d, Ω(M)) as well as (d, DN Ω(M)) on p p D0 p p D0 p L Ω(M) is dLp where D(dLp ) = {ω ∈ L Ω(M): d N ω ∈ L } = {ω ∈ L Ω(M): d ω ∈ L }. p By Hodge duality, the closure of (δ, Ω(M)) as well as (δ, DDΩ(M)) on L Ω(M) is δLp where D(δLp ) = p 0 p p 0 p {ω ∈ L Ω(M): δDD ω ∈ L } = {ω ∈ L Ω(M): δD ω ∈ L }.

58 ∗ ∗ p p p Define δc,L = dLp0 and dc,L = δLp0 . Then δc,L is the closure of (δ, DN Ω(M)) as well as (δ, DΩ(M)). 0 p DN p Also, D (δc,Lp ) = {ω ∈ L Ω(M): δc ω ∈ L }. p Similarly, dc,Lp is the closure of (d, DDΩ(M)) and (d, DΩ(M)). Also, D (dc,Lp ) = {ω ∈ L Ω(M): 0 DD p dc ω ∈ L }.

Lp Lp Proof. Firstly, it is trivial to check dLp is closed and (d, Ω(M)) is closable (ωj −−→ 0 and dωj −−→ η 0 D ε ε would imply η = 0 since dωj −−→ 0). Then let ω ∈ D(dLp ). We can conclude (ω , d (ω )) =   0 ε p p  0  ε D L ⊕L D ω , d N ω −−−−→ ω, d N ω . This also gives the closure of (d, DN Ω(M)). ε↓0 p p p0 p0 Then let G (δc,Lp ) ≤ L Ω ⊕ L Ω be the graph of δc,Lp . Similarly for G (dLp0 ) ≤ L Ω ⊕ L Ω. ⊥ Write J(x, y) = (−y, x). By the definition of adjoints, J (G (δc,Lp )) = G (dLp0 ) . Then observe that

⊥  p p  p0 p0 (L ⊕ L ) -cl {(−δcφ, φ): φ ∈ DΩ} = {(ω1, ω2) ∈ L ⊕ L : hhω1, δcφiiΛ = hhω2, φiiΛ ∀φ ∈ DΩ}

p0 p0 D0 = {(ω1, ω2) ∈ L ⊕ L : ω2 = d ω1} = G (dLp0 )

p p Then G (δc,Lp ) = (L ⊕ L ) -cl {(φ, δcφ): φ ∈ DΩ}. Do the same for φ ∈ DN Ω. Finally, by the definition of adjoints:

 p D (δc,Lp ) = ω ∈ L Ω(M): hhω, dLp0 φiiΛ . kφkLp0 ∀φ ∈ D (dLp0 )  0  p DN ε ε ε ε = ω ∈ L Ω(M): δc ω (φ ) = |hhω, dφ ii | = hhω, (d p0 φ) ii kφ k p0 ∀φ ∈ D (d p0 ) , ∀ε > 0 Λ L Λ . L L

 0  0 p DN p DN p = ω ∈ L Ω(M): δc ω (φ) kφk p0 ∀φ ∈ DN Ω = {ω ∈ L Ω(M): δc ω ∈ L } . L

C∞ t∆gN k For the third equal sign, we implicitly used the fact that e φ −−→ φ ∀φ ∈ D N Ω . t↓0

1,p 1,p 1,p In particular, W ΩN = W -cl (DN Ω) ≤ D (δc,Lp ). Similarly, W ΩD ≤ D (dc,Lp ). This makes our choice of notation consistent. Interestingly, a literature search yields a similar result regarding the adjoints of d and δ in [AM04, Proposition 4.3], where the authors used Lie flows on the domain M which is bounded in Rn, as well as n ∗ zero extensions to R to characterize D(dLp ) and D (dLp ). In [MM09a, Equation 2.12], for η ∈ D(δLp ), the 1  1 ∗ − p p authors defined ν ∨ η ∈ Bp,p Ω(∂M) = Bp0,p0 Ω(∂M) (p ∈ (1, ∞)) by

∗ DD D0 EE hhν ∨ η,  ωiiΛ = hhη, dωiiΛ − δ η, ω ∀ω ∈ Ω(M) Λ

∗ which is reminiscent of Equation (11). Note that hhν ∨ η,  ωiiΛ is abuse of notation (referring to the natural Trace 1 1,p0 1 p pairing via duality). Recall from Blackbox 47 that W Ω(M) = Fp0,2Ω(M)  Bp0,p0 Ω(M) ∂M has a ∗ bounded linear section Ext, so it is possible to choose ω such that k ωk 1 ∼ kωkW 1,p0 and therefore ν ∨ η B p p0,p0 is well-defined with

0 ∗ D kν ∨ ηk − 1 ∼ sup |hhν ∨ η,  ωiiΛ| . kηkLp + δ η B p 0 Lp p,p ω∈W 1,p Ω(M) ∗ k ωk 1/p =1 B p0,p0

59 1,p ∗ Of course, for η ∈ W Ω, ν ∨ η =  ιν η. We can now show an alternative description of D (δc,Lp ):

0 0 p DN p p D Theorem 80. For p ∈ (1, ∞), D (δc,Lp ) = {η ∈ L Ω(M): δc η ∈ L } = {η ∈ L Ω(M): δ η ∈ Lp and ν ∨ η = 0}.

0 0 0 p DN p p DN D Proof. Assume η ∈ L Ω(M) and δc η ∈ L . Then ∃α ∈ L Ω(M): α = δc η = δ η. By the 0 ∗ DN definition of ν ∨ η, hhα, ωiiΛ + hhν ∨ η,  ωiiΛ = hhη, dωiiΛ ∀ω ∈ Ω(M). By the definition of δc η, ∗ hhα, ωiiΛ = hhη, dωiiΛ ∀ω ∈ DN Ω. So hhν ∨ η,  ωiiΛ = 0 ∀ω ∈ DN Ω. Recall that Ext (the right 1 p ∗ Ext ∗ 1,p
∗ 1,p
Ext inverse of Trace) is bounded, so Bp0,p0 -cl ( (DN Ω)) =  W -cl (DN Ω) =  W ΩN (M) = 1 ∗ 1,p
p  W Ω(M) = Bp0,p0 Ω(∂M). Therefore ν ∨ η = 0. 0 Conversely, now assume η ∈ LpΩ(M), δD η = α ∈ Lp and ν ∨ η = 0. Then by the definition of

ν ∨ η for η ∈ D(δLp ), hhα, ωiiΛ = hhη, dωiiΛ ∀ω ∈ Ω(M). The formula also holds for ω ∈ DN Ω, and 0 DN p therefore δc η = α ∈ L .

This result agrees with [MM09a, Equation 2.17]. Our characterization of the adjoints of d and δ further highlights how heatable currents are truly natural objects in Hodge theory, independent of the theory of heat flows.   0 p p DN In particular, it is trivial to show L Ω = L -cl Ker δc = {η ∈ D (δc,Lp ): δc η = 0} for p ∈ (1, ∞). P ΩN Remark. The name “heatable current” simply refers to the largest topological vector space of differential forms (and hence vector fields) for which the heat equation can be solved (i.e. heatable), and once we apply the heat flow a heatable current becomes heated. The name “current” for distributional forms was introduced by Georges de Rham [Rha84], likely with its physical equivalents in mind, and has since become standard in various areas of mathematics such as geometric measure theory and complex manifolds. It is not easy to search for literature dealing with the subject and how it relates to Hodge theory. They are mentioned in a couple of papers [BB97; Tro09] dealing with “tempered currents” or “temperate currents” on Rn – differential forms with tempered-distributional coefficients. Yet the notion of “tempered” – not growing too fast – does not make sense on a compact manifold with boundary. Arguably, it is the ability to facilitate the heat flow, or the Littlewood-Paley projection, that most characterizes tempered distributions and makes them ideal for harmonic analysis. For scalar functions, much more is known (cf. [KP14; BBD18; Tan18] and their references). In the same vein, various results from harmonic analysis should also hold for heatable currents.

7.5 Square root We will not need this for the rest of the paper, but a popular question is the characterization of the square root of the Laplacian. N⊥ 1 k 1 By the Poincare inequality, P H ΩN is a Hilbert space where the H -inner product can be replaced by (ω, η) 7→ D(ω, η) (the Dirichlet integral). The space is dense in PN⊥L2Ωk. Define A as an unbounded operator on PN⊥L2Ωk where

N⊥ 1 k N⊥ 1 k D (A) = {ω ∈ P H ΩN : |D(ω, η)| .ω kηk2 ∀η ∈ P H ΩN }

N⊥ 1 k  −1  and hhAω, ηiiΛ = D(ω, η) ∀ω ∈ D(A), ∀η ∈ P H ΩN . Easy to check that hhAω, ηiiΛ = D (−∆N ) Aω, η N⊥ 1 k −1 N⊥ 2 k ∀η ∈ P H ΩN . Therefore ω = (−∆N ) Aω ∈ P H ΩhomN and Aω = (−∆N ) ω ∀ω ∈ D(A), so A ⊂ −∆N . It is trivial to check D (−∆N ) ≤ D (A), so A = −∆N .

60 By Friedrichs extension (cf. [Tay11a, Appendix A, Proposition 8.7], [Tay11b, Section 8, Proposition 2.2]), we conclude that h  i N⊥ 1 k N⊥ 2 k C
 N⊥ 2 k N⊥ 2 k  CP H ΩN = CP L Ω , D ∆N , k·kD ∆C = CP L Ω , CP H ΩhomN 1 ( N ) 1 2 2  q   = D −∆C , k·k √  N C D −∆N

By direct summing, we can extend the result to ∆gN to get

q  ! 1 k  2 k 2 k  C CH ΩN = CL Ω , CH ΩhomN 1 = D −∆gN , k·k q  2 C D −∆gN

∗ We note that the norms are only defined up to equivalent norms, and k·kD(A) is not the same as k·kD(A) (see Section2). This difference is not always made explicit in [Tay11a; Tay11b].

7.6 Some trace-zero results Although we will not need them for the rest of the paper, let us briefly delineate some results regarding the trace-zero Laplacian (cf. Theorem 61) which are similar to those obtained above for the absolute Neumann Laplacian. We begin by retracing our steps from Corollary 63. k k k k 0 Define H0 (M) = HN (M) ∩ HD (M). Obviously, H0 (M) is finite-dimensional and we can define P and P0⊥ the same way we did for PN and PN⊥ in Corollary 63. When M has no boundary, P0⊥ = PN⊥ and 0 N P = P = P3. It is a celebrated theorem, following from the Aronszajn continuation theorem [AKS62], that k H0 (M) = 0 when every connected component of M has nonempty boundary (cf. [Sch95, Theorem 3.4.4]). When that happens, P0⊥ = 1 and P0 = 0.

Blackbox 81 (Potential theory). For m ∈ N0, p ∈ (1, ∞), we define the injective trace-zero Laplacian

0⊥ m+2,p k 0⊥ m,p k ∆0 : P W Ω0 → P W Ω

−1 as simply ∆ under domain restriction. Then (−∆0) is called the trace-zero potential, which is bounded. 0⊥ m,p k ∆0 can also be thought of as an unbounded operator on P W Ω0 .

Proof. We only need to prove the theorem on each connected component of M. So WLOG, M is connected. If ∂M = ∅, we are back to the absolute Neumann case in Blackbox 64. When ∂M 6= ∅, 0⊥
P = 1 and we only need to show the trace-zero Poisson problem ∆ω, ω ∂M = (η, 0) is uniquely solvable for each η ∈ W m,pΩk. This is [Sch95, Theorem 3.4.10].

Consequently, we have a trivial decomposition

0⊥ 0 −1 0⊥ −1 0⊥ 0 ω = P ω + P ω = dδ (−∆0) P ω + δd (−∆0) P ω + P ω

m,p k for ω ∈ W Ω , m ∈ N0, p ∈ (1, ∞). This decomposition is not as useful as the Hodge-Morrey decompo- sition (Subsection 6.4) since the the first two terms are not orthogonal. However, it does mean that, when P0 = 0, every differential form is a sum of exact and coexact forms. 0⊥ m+2,p k −1 −1 For ω ∈ P W Ω0 , m ∈ N0, p ∈ (1, ∞), we also have ω = (−∆0) (−∆0) ω = (−∆0) (dδω + δdω) , so kωkW m+2,p ∼ kδωkW m+1,p + kdωkW m+1,p . This trick is not enough to get the full Poincare inequality

61 kωkW 1,p ∼ kδωkp + kdωkp, and therefore [Sch95, Lemma 2.4.10.iv] might be wrong. −1 0⊥ 2 k As (−∆0) is symmetric and bounded on P L Ω , we conclude ∆0 is a self-adjoint and dissipative 0⊥ 2 k 0⊥ 2 k C operator on P L Ω , with the domain D (∆0) = P H Ω0 . This means ∆0 is acutely sectorial on CP0⊥L2Ωk. 2 k Next we define the non-injective trace-zero Laplacian ∆f0 as an unbounded operator on L Ω with  m m m k m C 2 k D ∆f0 = D (∆0 ) ⊕ H0 and ∆f0 = ∆0 ⊕ 0 ∀m ∈ N1. Again, ∆f0 is acutely sectorial on CL Ω and m     0⊥ 2 k k 2 k kωk m ∼ kωk 2m ∀ω ∈ D ∆0 , ∀m ∈ 1. In particular, D ∆0 = P H Ω ⊕ H = H Ω . D(∆f0 ) H f N f 0 0 0 p 2
2 2,p k For L -analyticity, observe that on ∂M: ∇ν |ω| = 2 |h∇ν ω, ωi| = 0 . |ω| ∀ω ∈ W Ω0 , ∀p ∈ (1, ∞). So we argue as in Theorem 71, and Lp-analyticity follows. 2 k 2 k Remark. The operator P∆f0, with the domain H Ω0 ∩ PL Ω , is a well-defined unbounded operator on PL2Ωk. It is called the Stokes operator corresponding to the trace-zero/no-slip boundary condition, as discussed in [FK64; GM85; MM08] and others. It lies outside the scope of this paper. For more information, see [HS18] and its references.

8 Results related to the Euler equation

8.1 Hodge-Sobolev spaces We will have need of negative-order Sobolev spaces when we calculate the pressure in the Euler equation. 0 N⊥ Recall the space of heatable currents DN Ω (defined in Subsection 7.4). Note that P is well-defined on 0 N⊥ N⊥ 0 N DN Ω by P Λ, φ Λ = Λ, P φ ∀Λ ∈ DN Ω, ∀φ ∈ DN Ω. Same for P , and we can uniquely identify N 0 P Λ ∈ HN ∀Λ ∈ DN Ω. ex N Similarly, P (DN Ω) ≤ DN Ω (use Theorem 76 and Theorem 68), so P, 1−P = (P1 + P3 ) and P2 = P−P 0 are well-defined on DN Ω. 0 D0 0 D0 DN N N⊥ 0 DN N 0 For all p ∈ (1, ∞), define DN = d +δc on P DN Ω and DgN = d +δc on DN Ω as the injective and non-injective (Neumann) Hodge-Dirac operators. N⊥ N⊥ By the Poincare inequality (Corollary 69), it is easy to check that DN N⊥ : P D N Ω → P D N Ω P DN Ω N⊥ 0 is bijective. Consequently, so is DN on P DN Ω. N⊥ m,p −1 N⊥ m+1,p Observe that ∀m ∈ N0, ∀p ∈ (1, ∞) , ∀α ∈ P W Ω(M) , ∃!β = (DN ) α ∈ P W ΩN and

kβkW m+1,p ∼ kαkW m,p = kdβ + δcβkW m,p ∼ kdβkW m,p + kδcβkW m,p (12)

N⊥ m,p m+1,p
m+1,p
because P W Ω = d W Ω ⊕ δc W ΩN is a direct sum of closed subspaces (corresponding ex to P1 + P3 and P2). 0 0 0 0 D0 0 DN DN DN 2 2 DN N DN Note that we do not have d DN = DN d , but d DN = DN d = −∆N d is true.

m,p −m N⊥ p
N⊥ 0 Definition 82. For m ∈ Z, p ∈ (1, ∞), let W (DN ) := (DN ) P L Ω = {α ∈ P DN Ω: m p m,p   m,p (DN ) α ∈ L Ω} and W DgN := W (DN ) ⊕ HN . They are Banach spaces under the norms m N⊥ N kαk m,p := k(DN ) αk p and kβk m,p := P β m,p + P β . W (DN ) L Ω W (DgN ) W (DN ) HN In a sense, these are comparable to homogeneous and inhomogeneous Bessel potential spaces. We can extend the definitions to fractional powers, but that is outside the scope of this paper.

It is trivial to check that kαk m,p ∼ kαk m,p ∀α ∈ DN Ω, ∀m ∈ 0, ∀p ∈ (1, ∞). W (DgN ) W Ω N

m,p   Theorem 83. Some basic properties of W DgN :

62 m,p   1. DN Ω is dense in W DgN ∀m ∈ Z, ∀p ∈ (1, ∞).

m,p   m,p 2. W DgN = W -cl (DN Ω) ∀m ∈ N0, ∀p ∈ (1, ∞).

0 D0   DN N m+1,p 3. d β + δc β kβk m+1,p ∀β ∈ W DgN , ∀m ∈ , ∀p ∈ (1, ∞) m,p m,p . W (DgN ) Z W (DgN ) W (DgN ) −1 −1 D0 0  D0  D0  D0  0 N DN N N⊥ N N DN N Then P2 = δc d −∆N P = δc −∆N d and P = P2 + P are of order 0 on m,p   W DgN .

∗  m,p   −m,p0   4. W DN = W DN ∀m ∈ , ∀p ∈ (1, ∞) via the pairing hα, φi −m,p m,p0 = g g Z W (DgN ),W (DgN ) −m N⊥ m N⊥ N N DN P α, DN P φ Λ + P α, P φ Λ

Proof.

m N⊥
N⊥ N⊥ p 1. Because DN P DN Ω = P DN Ω is dense in P L Ω.

m,p N⊥ m,p   m,p N⊥
N⊥ m,p   2. We only need W (DN ) = P W DgN ≤ W -cl P DN Ω . Let α ∈ P W DgN Lp W m,p ε ε∆gN m ε m ε m −m m ε ε and α = e α as usual. Then DN (α ) = (DN α) −−→ DN α. So DN DN (α ) = α −−−−→ ε↓0 ε↓0 α by Equation (12).

m+1 N⊥ p m N⊥ N⊥ 1,p 3. Let DN P β ∈ L . Then DN P β ∈ P W ΩN by Equation (12).

2k N⊥ 2k N⊥ 2k N⊥ 2k N⊥ When m = 2k (k ∈ Z): dDN P β Lp + δcDN P β Lp ∼ dDN P β + δcDN P β Lp = 2k+1 N⊥ DN P β Lp .

2k N⊥ N⊥ 2,p When m = 2k + 1 (k ∈ Z): DN P β ∈ P W Ωhom N and

2k N⊥ 2k N⊥ 2k N⊥ 2k N⊥ DN dDN P β Lp + DN δcDN P β Lp = δcdDN P β Lp + dδcDN P β Lp 2k N⊥ 2k N⊥ 2k+2 N⊥ ∼ δcdDN P β + dδcDN P β Lp = DN P β Lp

m DN m,p ∗ −m,p0 m,p ∼ 4. Simply observe that (W (DN )) = W (DN ) via the isomorphisms W (DN ) −→ −m DN N⊥ p −m,p0 ∼ N⊥ p0 P L Ω and W (DN ) −→ P L Ω.

N⊥ 1 N⊥ 2 Remark. We briefly note that DN with the domain P H ΩN is self-adjoint on P L Ω and its complex- ification is therefore “bisectorial”. For more on this, see [McI86; McI10; MM18].

2 0 Corollary 84. Assume U ∈ PL X. Define div(U ⊗ U) ∈ DN X by hhdiv(U ⊗ U),XiiΛ := − hhU ⊗ U, ∇Xii ∀X ∈ DN X. p [ If p ∈ (1, ∞) and U ⊗ U ∈ L Γ(TM ⊗ TM), then div(U ⊗ U) −1,p kU ⊗ Uk p . W (DgN ) . L

k Proof. For η ∈ Ω(M) , write ηk for the part of η in Ω . Let φ ∈ DN Ω, then

DD −1 N⊥ [ EE DD −1 N⊥
] EE 0 DN P div(U ⊗ U) , φ = U ⊗ U, ∇ DN P φ . kU ⊗ UkLp kφkLp Λ 1

63   DD EE This implies div(U⊗U)[ ∈ W −1,p D . Then observe div(U ⊗ U)[, φ = U ⊗ U, ∇ (φ)] gN Λ 1 .

kU ⊗ Uk p kφk 1,p0 . L W (DgN )

8.2 Calculating the pressure

0 X DN (I, ) 2 2
1 In this subsection, we assume that ∂tV + div(V ⊗ V) + grad p == 0, V ∈ Lloc I, PL X , p ∈ Lloc(I × M). This is true, for instance, in the case of Onsager’s conjecture (see Subsection 3.3 and Subsection 3.4). 0 0 We first note that HN = H = {locally constant functions}. Then we can show V uniquely determines 0 N⊥ p by a formula, up to a difference in HN (dp is always unique). It is no loss of generality to set p = P p R (implying M p = 0).

q m+1,p 1. Assume V ⊗ V ∈ Lt W Γ(TM ⊗ TM) for some m ∈ N0, p ∈ (1, ∞) , q ∈ [1, ∞]. 0 0 (I,X) [ D DN q m,p 1 Let ω = div(V ⊗ V) . Then d N p == (P − 1) ω ∈ Lt W Ω . By the Poincare inequality (Corol- 0 (I,X) 0 q N⊥ m+1,p 0 DN D lary 69), there is a unique f ∈ Lt P W Ω such that df = (P − 1) ω == d N p. An explicit N⊥ −1 D⊥ N⊥ −1 ex formula is f = −Rdω where Rd := P δ (−∆D) P + P δ (−∆N ) P3 is the potential for d. ∞ 0
0 0
N 0
We aim to show f = p. Let ψ ∈ Cc I, DN Ω . Then because Ω = P2 Ω ⊕ P3 Ω , we conclude N⊥ −1 N⊥ ∞ 1
P ψ = δcφ where φ := d (−∆N ) P ψ ∈ Cc I, DN Ω and

Z Z Z Z Z DD 0 EE Z N⊥ DN hhf, ψiiΛ = f, P ψ Λ = hhf, δcφiiΛ = hhdf, φiiΛ = d p, φ = hhp, ψiiΛ I I I I I Λ I

Therefore p = f and kpk q m+1,p kωk q m,p kV ⊗ Vk q m+1,p . Lt W . Lt W . Lt W

q p 2. Assume V ⊗ V ∈ Lt L Γ(TM ⊗ TM) for some p ∈ (1, ∞) , q ∈ [1, ∞]. 0 0 (I,X)   [ D DN q −1,p Let ω = div(V ⊗V) . Then d N p == (P − 1) ω ∈ Lt W DgN by Corollary 84 and Theorem 83. 0 0 0 0 0 D 0 DN (I,X) D D q   D N DN N N −2,p −2 N Then −δc d p == δc (1 − ) ω = δc ω ∈ L W DgN and p = −D δc ω, so kpk q p P t N Lt L . 0 DN δc ω kωk q −1,p kV ⊗ Vk q p . q −2,p . Lt W (DgN ) . Lt L Lt W (DgN )

−1 0  D0  DN N N⊥ 0 −1
N⊥ Remark. It is also possible to define Rδc := d −∆N P on DN Ω and have Rd = DN − Rδc P 0 m,p   on D Ω. This would then imply kRdαk m+1,p kαk m,p ∀α ∈ W DN , ∀m ∈ , ∀p ∈ N W (DgN ) . W (DgN ) g Z (1, ∞).

8.3 On an interpolation identity

1 p Let p ∈ (1, ∞). We are faced with the difficulty of finding a good interpolation characterization for Bp,1ΩN . 1 p p 1,p
We do have Bp,1Ω = L Ω,W Ω 1 (complexification, then projection onto the real part), but our heat p ,1 1 1,p p p 1,p
flow is not analytic on CW Ω. The hope is that Bp,1ΩN = L Ω,W ΩN 1 , and our first guess is to try p ,1 to find some kind of projection. Indeed, the Leray projection yields

1 p p 1,p
PBp,1Ω = PL Ω, PW Ω 1 (13) p ,1

64 1,p 1,p and the heat flow is well-behaved on PW Ω = PW ΩN (Theorem 68, Theorem 76). By interpolation, P 1 1 1 1 p p p p p is Bp,1-continuous, so nP : Bp,1Ω → L Ω ∂M is continuous and PBp,1Ω = PBp,1ΩN . This is enough to get all the Besov estimates we will need for Onsager’s conjecture. 1 p Additionally, it is true that the heat semigroup is also C0 and analytic on CPBp,1ΩN by Yosida’s half- plane criterion (Theorem 39). Unlike the Lp-analyticity case, here we already have analyticity on the 2 endpoints, so the criterion simply follows by interpolation. Alternatively, observe that there exists C > 0     t ∆gC −C C N p 1,p such that supt>0 t ∆gN − C e < ∞ for V ∈ {CPL Ω, CPW ΩN }. Therefore it also holds L(V ) 1 p for V = CPBp,1ΩN by interpolation, and that is another criterion for analyticity ([Eng00, Section II, Theorem 4.6.c]). 1 p 1,p
p Unfortunately, this does not tell us about the relationship between L Ω,W ΩN 1 and Bp,1ΩN . p ,1 1 p 1,p
p 1,p Obviously L Ω,W ΩN 1 ,→ Bp,1ΩN by the density of W ΩN . The other direction is more delicate. p ,1 Interpolation involving boundary conditions is often nontrivial. The reader can see [Gui91; Lof92; Ama19] 1 to get an idea of the challenges involved, especially at the critical regularity levels N + p . Nevertheless, there are a few interesting things we can say about these spaces.

Definition 85 (Neumann condition on strip). For vector field X and r > 0 small, with ψr as in Equation (3), define nrX = ψr hX, νei νe and trX = X − nrX m,p s Then define XN,r = {X ∈ X : hX, νei = 0 on M

Some basic facts:

1. t X ≤ X r r N, 2

2. tr = 1 and nr = 0 on XN,r

3. t r t = t 2 r r

4. ktrXkW m,p .r,m,p kXkW m,p for m ∈ N0, p ∈ [1, ∞]

m,p s 5. W XN,r and Bp,qXN,r are Banach for m ∈ N0, p ∈ [1, ∞], s ≥ 0, q ∈ [1, ∞]

m tr =1 m ,p m ,p
m 6. B θ X ,−−−→ W 0 X r ,W 1 X r ,→ B θ X r for θ ∈ (0, 1) , m ∈ , m 6= m , p ∈ p,q N,r N, 2 N, 2 θ,q p,q N, 2 j N0 0 1 [1, ∞], q ∈ [1, ∞], mθ = (1 − θ) m0 + θm1.

Remark. The last assertion is proven by the definition of the J-method, and it works like partial interpolation.

The reader can notice the similarity with the Littlewood-Paley projection (P≤N P N = P N ). The hope is ≤ 2 ≤ 2 t↓0 that trX −−→ X in a good way for X ∈ XN . m θ   m A subtle issue is that for X ∈ Bp,q XN,r, kXk m ,p m ,p .r kXkB θ X . The implicit W 0 XN, r ,W 1 XN, r p,q N,r 2 2 θ,q constant which depends on r can blow up as r ↓ 0. s s s
m,p m,p m,p Define Bp,qXN,0+ = Bp,q-cl ∪r>0 smallBp,qXN,r and W XN,0+ = W -cl (∪r>0 smallW XN,r). Then we recover the usual spaces by results from Subsection 5.5:

Theorem 86. Let p ∈ (1, ∞):

65 p p 1,p 1,p 1. L XN,0+ = L X,W XN,0+ = W XN .

1 1 p p 2. Bp,1XN,0+ = Bp,1XN .

Proof.

p p L 1,p 1. Let X ∈ L X. Then nrX −−→ 0 by shrinking support. If X ∈ W XN , then by Theorem 53 r↓0

knrXk 1,p = kψr hX, νik 1,p kψrk 1,∞ khX, νik p + kψrk ∞ khX, νik 1,p W e W (M

1 1 p p ∞ 1,∞
1,∞ 2. Let Y ∈ B X. As B∞,∞ = L ,W 1 and ψ ∈ W , we conclude kψ k 1 p,1 ,∞ r r p . p B∞,∞ 1 1 1 p0 p 1
p kψrkL∞ kψrkW 1,∞ . r . Then by Theorem 53 and Theorem 54:

1 1 p knrY k 1 ¬r kψrk 1 khY, νik p + kψrk ∞ khY, νik 1 khY, νik p + kY k 1 p . p e L (M<4r ) L e p . e L (M<4r ) p Bp,1 B∞,1(M) Bp,1(M) r Bp,1

khY, νik p + kY k 1 ¬r kY k 1 . e L (M<4r ,avg) p . p Bp,1(M) Bp,1

Therefore knrY k 1/p does not blow up as r ↓ 0. Then we make a dense convergence argument: Bp,1 1 B1/p p p,1 j→∞ assume X ∈ Bp,1XN and let Xj ∈ X such that Xj −−−→ X, then khXj, νikLp(∂M) −−−→ 0. Note that we do not have nXj = 0. By Theorem 53:

1 1 p0 p kn X k 1 kn X k p kn X k 1,p r j p . r j L r j W Bp,1 1  1 1 1 1  p0 p p p p khXj, νik p kψrk 1,∞ khXj, νik p + kψrk ∞ khXj, νik 1,p . e L (M

1   p 1 1 1 p0 p khXj, νik p + khXj, νik p khXj, νik 1,p . e L (M

So lim sup kn X k 1 khX , νik p and r↓0 r j p . j L (∂M) Bp,1

lim sup kn Xk 1 lim sup kn (X − X )k 1 + lim sup kn X k 1 r p . r j p r j p r↓0 Bp,1 r↓0 Bp,1 r↓0 Bp,1

kX − X k 1 + khX , νik p . j p j L (∂M) Bp,1

As j is arbitrary, let j → ∞ and lim sup kn Xk 1 = 0. r↓0 r p Bp,1

These results hold not just for vector fields, but also for differential forms once we perform the proper modifications: for differential form ω, define n ω = ψ ν[ ∧ (ι ω), t ω = ω − n ω, W m,pΩk = {ω ∈ W m,pΩk : r r e νe r r r

66 1 1 ι ω = 0 on M }, replace hX, νi with ι ω in the proofs etc. In particular, B p Ωk = B p Ωk for νe

A Complexification

Throughout this small subsection, the overline always stands for conjugation, and not topological closure.  φ  Let RX be a real NVS, then a complexification of RX is a tuple CX, RX ,−→ CX such that

1. CX is a complex NVS.

2. φ is a linear, continuous injection and φ(RX) ⊕ iφ(RX) = CX. 3. kφ(x)k = kxk and kφ(x) + iφ(y)k = kφ(x) − iφ(y)k ∀x, y ∈ X. CX RX CX CX R The last property says k·k is a complexification norm. By treating φ( X) as the real part, ∀z ∈ X, CX R C we can define

Construction A standard construction of such a complexification is CX = RX ⊗R C. As RX is a flat and = φ = free R-module, 0 → R ,→ C  R → 0 induces 0 → RX ,−→ CX  RX → 0 as a split short exact sequence and CX = φ(RX) ⊕ iφ(RX). Then we can make φ implicit and not write it again. The representation z = x + iy = (x, y) is unique. Easy to see that any two complexifications of RX must be isomorphic as C-modules. We define the minimal complexification norm (also called Taylor norm)

kx + iyk := sup kx cos θ − y sin θk = sup

Any other complexification norm is equivalent to k·kT .

Proof. Let k·k be another complexification norm. Then

So the topology of CX is unique. It is more convenient, however, to set kx + iyk X = k(x, y)k X⊕ X = 1 C R R   2 kxk2 + kyk2 ∀x, y ∈ X. Easy to see that any two complexifications of X must be isomorphic as RX RX R R

complex NVS, so we write CX = RX ⊗R C from this point on, and if RX is normed, so is CX. Obviously, if RX is Banach, so is CX, and when that happens, we call (RX, CX) a Banach complexification couple.

Real operators Let (RX, CX) and (RY, CY ) be 2 Banach complexification couples.

• An operator A : D (A) ≤ CX → CY is called a real operator when D (A) = C

• An unbounded R-linear operator T : D(T ) ≤ RX → RY has a natural complexified version T C = C
C T ⊗R 1C : CX → CY where D T = CD (T ). Obviously T is a real operator and we write (T,T C) (RX, CX) −−−−→ (RY, CY ).

67
– D (T C) = D T C and T Cz = T Cz ∀z ∈ CX. – T is closed ⇐⇒ T C is closed. Same for bounded, compact, densely defined.

• For any unbounded C-linear operator A : D (A) ≤ CX → CY such that D (A) = C< (D (A)), define 2    C 

(T,T C) Spectrum For (RX, CX) −−−−→ (RY, CY ), define • ρ(T ) := ρ T C
, σ(T ) := σ T C
.

• ρR(T ) := {λ ∈ R : λ − T is boundedly invertible} and σR(T ) := R\ρR(T ). If ζ ∈ C and ζ − T C is boundedly invertible, so is ζ − T C. So σ (T ) = σ (T ) and ρ(T ) = ρ(T ). C For λ ∈ R, λ − T is boundedly invertible ⇐⇒ λ − T is boundedly invertible. So ρR(T ) = ρ(T ) ∩ R and σR (T ) = σ (T ) ∩ R.

Semigroup T generates an R-linear C0 semigroup ⇐⇒ T C generates a C-linear C0 semigroup. When that happens, etT
C = etT C .

Proof. When either happens, T and T C are densely defined. Also, T − j and T C − j are boundedly invertible for j ∈ N large enough, so T and T C are closed. Easy to use Hille-Yosida to show both T and T C must generate C0 semigroups. 1 1 As in the proof of Hille-Yosida, define the Yosida approximations T = T 1 , T C = T C 1 = j j C 1− j T 1− j T C C C C tTj
C tTj tT
C tT (Tj) . As Tj and Tj are bounded, e = e by power series expansion. Then e = e tT tTj as e = limj→∞ e pointwise.

Hilbert spaces Let RH be a real Hilbert space with inner product h·, ·i. Then CH is also Hilbert with the inner product

hx1 + iy1, x2 + iy2i := hx1, x2i + hy1, y2i + i (hy1, x2i − hx1, y2i) ∀xj, yj ∈ H CH R

1   2 Then kx + iyk = kxk2 + kyk2 ∀x, y ∈ H, consistent with our previously chosen norm. CH RH RH R

Also, hz1, z2i H = hz2, z1i H ∀z1, z2 ∈ CH.
C C Let A, AC :(RH, CH) → (RH, CH) be unbounded. • A is symmetric ⇐⇒ AC is symmetric. When that happens, hAx + iAy, x + iyi = hAx, xi + CH hAy, yi ∀x, y ∈ RH.
 ⊥
⊥ • C (RH ⊕ RH) = CH ⊕ CH and G AC = CG (A) (graphs). Also C G (A) = G AC .

• A is self-adjoint ⇐⇒ AC is self-adjoint. When this happens, σ(A) = σ(AC) ⊂ R. • A is dissipative ⇐⇒ AC is dissipative. For more information on complexification, see [Gl¨u17, Appendix C].

68 Nomenclature

ψr, fr cutoffs on M living near the boundary, page 13

et∆ the absolute Neumann heat flow, defined for the proof of Onsager’s conjecture, page 14

L ((X0,X0) , (Y0,Y1)) morphisms between interpolation couples, page 20

(X0,X1)θ,q real interpolation, page 21

[X0,X1]θ complex interpolation, page 21 W m,p Sobolev spaces, page 30

s Bp,q Besov spaces, page 30

s Fp,q Triebel-Lizorkin spaces, page 30 Cs(Ω) Zygmund spaces, page 33

Ω

νe extension of ν near ∂M, page 37  : ∂M ,→ M is the smooth inclusion map, page 37

ι interior product (contraction) of differential forms, page 37 vol∂ volume form of ∂M, page 37

Γ(F), Γc(F), Γ00(F) the space of smooth sections of F with different support conditions, page 38 hhσ, θii dot product on Γ(F), page 38

RW m,p, CW m,p real and complexified versions of function space, page 40 XM set of smooth vector fields on M, page 40 t tangential part, page 40 n normal part, page 40

Ωk (M) set of smooth differential forms on M, page 41

[ ] Xp, ωp musical isomorphism, page 41 ? Hodge star, page 41

δ codifferential, page 41

∆ Hodge Laplacian, page 41

Rabcd Riemann curvature tensor, page 41

Ric Weitzenbock curvature operator, page 41

69 hT,Qi tensor inner product, page 42

hω, ηiΛ , hhω, ηiiΛ Hodge inner product, page 42 D(ω, η) Dirichlet integral, page 43

k k ΩD, ΩhomD different Dirichlet conditions for differential forms, page 44

k k ΩN , Ωhom N different Neumann conditions for differential forms, page 44

k k k H , HD, HN harmonic fields, then with Dirichlet and Neumann conditions, page 44 L2-cl (·) closure under L2 norm, page 44

PN , PN⊥, PD, PD⊥ natural orthogonal decomposition, page 45

∆N injective Neumann Laplacian, page 45

−1 −1 (−∆N ) , (−∆D) Neumann and Dirichlet potentials, page 45

δc, dc adjoints of d and δ, page 46

P1ω, P2ω, P3ω the component projections in Hodge decomposition, page 47

N ex D co P3 , P3 , P3 , P3 Friedrichs decomposition, page 49

P Leray projection, page 49

∆gN non-injective Neumann Laplacian, page 52

p Ap generator of heat flow on L , page 53

1,p A1,p generator of heat flow on W , page 55

k 0 k DN Ω , DN Ω heated forms and heatable currents, page 56

DN , DgN the injective and non-injective (Neumann) Hodge-Dirac operators, page 65

m,p m,p   W (DN ), W DgN Hodge-Sobolev spaces, page 65

CY,T C complexification of spaces and operators , page 68

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