TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 4, Pages 1297–1364 S 0002-9947(02)03214-2 Article electronically published on December 5, 2002

BESOV-MORREY SPACES: FUNCTION SPACE THEORY AND APPLICATIONS TO NON-LINEAR PDE

ANNA L. MAZZUCATO

Abstract. This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi- linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch’s results with those of Kozono and Ya- mazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo- differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

1. Introduction The focus of this work is the analysis of a class of function spaces, recently introduced by H. Kozono and M. Yamazaki to obtain critical regularity for the Navier-Stokes equation, and to study its applications to partial differential equa- tions. They are modeled on Besov spaces, but the underlying norm is of Morrey type rather than Lp (Definition 2.2). Throughout we will call them Besov-Morrey or BM spaces for short.1 As we will show in the paper, BM spaces share many of the properties of Besov spaces, but they represent local oscillations and singularities of functions more precisely. It follows their interest in the theory of partial differential equations, especially non-linear.

Received by the editors September 24, 2001 and, in revised form, September 9, 2002. 2000 Mathematics Subject Classification. Primary 35S05, 42B35; Secondary 76B03, 42C40, 35K55. Key words and phrases. Navier-Stokes equation, function spaces, Littlewood-Paley, pseudo- differential calculus, global analysis. This work was completed while the author was on leave from Yale University and visiting the Mathematical Sciences Research Institute (MSRI). Research at MSRI was supported in part by NSF grant DMS-9810361. 1An alternative name would be KY spaces, a convention followed in [Maz1].

c 2002 American Mathematical Society

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We start by recalling some notions of fluid dynamics, from which the definition of BM spaces arises quite naturally. The Navier-Stokes system (NS) on Rn in the incompressible, homogeneous, isotropic case is the following set of equations:

(1.1a) ∂tu(t, x) ν∆u(t, x)+u(t, x) u(t, x)+ p(t, x)=f(t, x), − ·∇ ∇ (1.1b) div u(t, x)=0,

where the vector field u(t, x) represents the velocity of the fluid at the position x and time t, ν is the (constant) viscosity coefficient, the scalar field p(x, t)isthe pressure, f(x, t) are external body forces. ν and f are given, while u and p are unknowns. The system must be complemented with an initial condition u(0,x)=a(x)and some prescribed behavior at infinity to obtain a possibly unique solution. In many applications, the forcing f admits a potential, which can be included in the pressure term. The pressure appears in the equations only to enforce the incompressibility condition (1.1b). So, it is customary to apply a projection onto the divergence-free vector fields, the so-called Leray projection P,onbothsidesof (1.1a), and study the (formally) simpler equation

(1.2) ∂tu(t, x) ν∆u(t, x)+P div(u u)(t, x)=0. − ⊗ The pressure p is then determined by solving a Poisson equation. Our concern here is with the behavior at fixed viscosity; so ν is conventionally set equal to one. It is a classical result (cf. [Tay1], [Tem]) that short-time, strong, unique solutions to (1.2) exist for sufficiently smooth initial data, e.g., for a in the Sobolev space Hk, k>n/2 + 1. On the other hand, in the 1930s, J. Leray [Lr] proved the existence of global weak solutions of finite energy (i.e., u L2), called Leray-Hopf solutions. Uniqueness and full regularity of these solutions∈ is still an open problem. But, if a strong solution exists, then the weak solution coincides with it. Therefore, it is relevant to prove existence and uniqueness under fairly weak assumptions on the initial data. Mild solutions can be obtained by rewriting (1.2) as the integral equation

t t∆ (t s)∆ (1.3) u(t, x)=e a(x) e − P div(u u)(s, x) ds , − ⊗ Z0 and then proving that, in suitable function spaces, the right-hand side defines a contraction. Here a does not necessarily have finite energy—its vorticity can be a measure; for instance, see [GM], [Ka2], [Tay2]. This last situation is important for the vortex ring and vortex sheet problem, where the vorticity ω =curlu is respectively a bounded measure supported on a closed curve or surface in R3. One of the most comprehensive studies in this context is that of M. Cannone, who in his Ph.D. thesis [Can] (1995) defined the notion of spaces adapted to NS. These are functional Banach spaces E with the additional property

(1.4) ψj(D)(fg) E ηj f E g E , j Z . k k ≤ k k k k ∀ ∈ ψj(D) is the Fourier multiplier with symbol ψj(ξ), where ϕ0(ξ),ψj(ξ) is a Little- { } wood-Paley partition of unity subordinate to a dyadic decomposition of Rn.The

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numbers ηj are such that

j (1.5) 2−| | ηj < . ∞ j X The above conditions ensure that the right-hand-side of (1.3) defines a Lipschitz map locally-in-time and, hence, that a short-time, unique solution to NS exists. Examples of adapted spaces include: (1) Xp(Rn)=P(Lp(Rn)) for p (n, ), p n ∈ ∞ (2) P( q(R )) for p (n, ), M ∈ ∞ p where q is the homogeneous Morrey space (Definition 2.1). Furthermore, every functionalM Banach space that is also an algebra is adapted, since (1.5) holds with ηj constant. In general, (1.4) and (1.5) are not enough to give long-time solutions. However, NS is invariant under an appropriate rescaling of the variables. So, when the norm of the space E also has some scaling behavior, it is quite reasonable to look for global existence by means of time dilations, for example, in the form of self-similar solutions. In this spirit, G. Karch [Kr] extended Cannone’s work by constructing for each E some auxiliary spaces of distributions:

α α/2 t∆ (1.6) BE = f 0 f BEα =sup t e f E < ,α 0 . { ∈S |k k t>0{ k k } ∞} ≥ He was able to prove long-time existence for sufficiently small initial data, if E has scaling degree n/q and α =1 n/q (so q n). The solution thus obtained is unique only in a− suitable subspace− of BEα. The≥ advantage of using the space BEα over E is that the norm of the initial data can be small in BEα without being so in p α α E. As a matter of fact, if E = L ,thenBE is the homogeneous Besov space Bp,− , where certain highly oscillatory functions exhibit such behavior [Can]. However,∞ in general, these solutions may not have finite energy. In 1993, H. Kozono and M. Yamazaki [KY] introduced two new classes of spaces, s s p,q,r and Np,q,r, which, as mentioned above, will be called homogeneous and in- Nhomogeneous Besov-Morrey spaces. s Np,q,r is the space of all tempered distributions such that

1/r sj r (1.7) f N s = ϕ0(D)f p + 2 ψj(D)f p < , k k p,q,r k kMq  k kMq  ∞ j 0 X≥
 while s is the space of all tempered distributions modulo polynomials such that Np,q,r   1/r sj r s p f = 2 ψj(D)f q (1.8) k kNp,q,r  k kM  j X∈Z
 sj p = 2 ψj(D)f q j∞= r < ,  { k kM } −∞ ` ∞

where p q n/p n/q p q (1.9) q f Lloc f q =sup sup R − f L (B(x0,R)) < , n M ≡{ ∈ |k kM x0 0

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and p q n/p n/q p q (1.10) Mq f Lloc f Mq =sup sup R − f L (B(x0,R)) < , n ≡{ ∈ |k k x0 0weak topology. We now give a brief outline of the paper. ∗ In Section 2, we introduce the BM and BEα spaces, and we compare them. s s Section 2.1 treats p,q,r and Np,q,r, starting with an overview of the theory of Morrey spaces. AsN expected, most of the features exhibited by BM spaces are analogous to those of Besov spaces. For example, they can be described as real p p interpolation of powers of the Laplacian acting on Mq and q.Inthiscontext,we complement the results in [KY] with some non-interpolationM results (Proposition 2.7).

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In view of applications to differential equations, it is important to measure regu- s s larity of distributions in p,q,r and Np,q,r. In [KY], Kozono and Yamazaki establish a Sobolev-type embeddingN for BM spaces:

s s n/p Np,q,r , C − , (1.11) → ∗ s ˙ s n/p p,q,r , C − , N → ∗ for s>n/p,whereCs and C˙ s are respectively inhomogeneous and homogeneous H¨older-Zygmund spaces.∗ 2 For ∗s = n/p,wehave n/p ˙ 0 p,q, C , N ∞ ⊂ ∗ (1.12) n/p BMO,r [1, 2], Np,q,r ⊂ ∈ n/p L∞, Np,q,1 ⊂ BMO being the space of functions of bounded mean oscillation [JN]. The relation with BMO is also addressed in Section 2.2, where we develop a wavelet decompo- sition for BM spaces in Proposition 2.13 and Proposition 2.14. Following [FJ1], we use a set of smooth wavelets that are not compactly supported, but are “concen- trated” on dyadic cubes. In particular, at the critical scaling for NS, Besov-Morrey 1 spaces are not comparable with the space BMO− of Koch and Tataru [KoT]. Although wavelets have not led to substantial progress in studying Navier-Stokes, they have proved useful for other (non-parabolic) equations. We are guided here, in particular, by work of M. Vishik [Vi2] on the Euler equation. Section 2.3 deals with Karch’s construction. In Proposition 2.22 we show that BEα can be identified with a homogeneous Besov-type space of negative index, based on E, if again the norm of E has some scaling degree. The intuitive reason t ξ 2 behind this result is that the symbol of the heat operator e− | | behaves like a bump function centered at the origin. By comparison, the norm of u in a Besov j space is defined in terms of smooth multipliers ψj(D)=ψ(2− D), where each function ψj(ξ) is supported on dyadic shells in frequency space. As a matter of fact, for classical Besov spaces it is a well-known result in interpolation theory that an equivalent norm can be obtained in terms of convolutions with the heat kernel [Trieb1]. However, our proof is based solely on rescaling arguments and it does not assume that the heat semigroup is strongly continuous on E. As a by-product, s s we obtain an equivalent characterization of p,q, and Np,q, in terms of thermic functions: N ∞ ∞ s/2 t∆ s p f p,q, sup t− e f q , k kN ∞ ≈ 0

2Recall that Cs Cs, the usual H¨older space, when s>0 is not an integer [Trieb3]. ∗ ≡

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Note that, as far as Navier-Stokes on flat Euclidean space is concerned, Fourier multipliers are sufficient. The situation is quite different on manifolds. In Section m 3.2, we are then led to consider more general symbols, belonging to the class S1,δ. Borrowing techniques employed for Sobolev and H¨older spaces [Ma], [Ru2], [Bour], 0 we establish rather general results; namely, operators with symbols in the class S1,δ s are bounded on Np,q,r for all δ [0, 1], if s>0. This includes the exotic class 0 ∈ S1,1, which is relevant in the study of para-differential operators and non-linear equations. For δ<1, good multiplicative properties of the symbols allow us to lift the restriction s>0. In the case q>1, the same conclusion can be reached s directly, using the representation of Np,q,r as a real interpolation space of powers of p (I ∆)Mq (Proposition 3.14). Moreover, one is not restricted to smooth symbols, − ` 0 but can work with the class of non-regular symbols C S1,δ,aslongas`>s>0, which is the content of Theorem 3.17. Here, too, a stronger∗ result holds if δ<1by means of symbol smoothing (Corollary 3.20). s In order to treat operators on the homogeneous space p,q,r,adifferentclass of symbols must be chosen to take into account possible singularitiesN at the origin. But we do not consider this problem here. In Section 3.3, we address para-differential calculus. Para-differential operators were introduced by J.-M. Bony [Bo] to study optimal regularity of solutions to non-linear partial differential equations. Here, we use a clever expansion (due to Y. Meyer [Me]) of a non-linear map F (u), similar to a Taylor polynomial. One is left with a smooth remainder R(u) and a linear part MF (u, D)u,where,however, MF is not a differential operator. In fact, if u L∞, MF is a pseudo-differential operator of type (1, 1). Since the non-linear term∈ in the Navier-Stokes equation is quadratic, we specialize to F (u1,u2)=u1 u2 and prove Moser-type estimates (equation (3.54)). These combined with the Sobolev embedding (1.11) imply that s Np,q,r is a Banach algebra for s>n/p(Corollary 3.22). Finally, we compare Meyer’s construction to the paraproduct of J.-M. Bony (see again [Bo]), which is obtained by applying symbol smoothing directly to F .Various estimates are established here (Corollary 3.23) that again are very similar to those for Sobolev and H¨older spaces. In Section 4, we address the problem of defining the BM spaces on manifolds. s We only deal with the local space Np,q,r, q>1, as pseudo-differential estimates on BM and Morrey spaces [Tay3] are used. The key step is, of course, to replace the multipliers ψj(D) with appropriate analogs on a curved non-Euclidean geometry. Indeed, operators on manifolds correspond, in general, to equivalence classes of symbols. We look at the simplest case, namely compact, Riemannian manifolds M without boundary, but we plan to consider manifolds with (smooth) boundary in subsequent work, exploiting the analysis developed here. To construct the BM spaces on M, we proceeded along two directions. In Section 4.1, we use a partition of unity and coordinate charts (Definition 4.5), n s n as we show that diffeomorphisms of R preserve the topology of Np,q,r(R ). The main ingredient in the proof consists of applying Egorov’s Theorem to obtain a 1 complete asymptotic expansion of the symbol of χ∗− ψj(D) χ∗,forχ adiffeo- ◦ ◦ morphism of Rn linear outside some compact set, and in studying the behavior of each term in this expansion when j . →∞

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s In Section 4.2, the topology of Np,q,r(M) is defined in terms of covariant objects; namely, a family of functions of √ ∆oftheformψj(√ ∆), where ∆ is the Laplace operator on M. − − Again, we first reduce the problem to local coordinates. Then, in Theorem 4.7, we carefully compare the symbols of ψj(D)andψj(A), where A is the operator obtained by transporting the Laplacian on M to Rn via a coordinate chart. An asymptotic formula for the symbol of ψj(A) can be obtained in terms of the solution operator to the wave equation cos(t√ ∆), by using geometrical optics methods. We follow Schrader and Taylor [ST], although− our approach is rather similar to that of Seeger and Sogge for Besov spaces [SeSo]. However, it is necessary here to bound explicitly the remainder term at each step in the symbol expansion, uniformly in j, s since no reduction to smooth functions is possible. Finally, Np,q,r(M) can still be p characterized as a real interpolation space of powers of (I ∆) acting on Mq (M) (Proposition 4.12). − Section 5 is finally devoted to the study of solutions to certain semi-linear par- abolic equations, with an emphasis on Navier-Stokes. Existence and regularity results for this class of equations were obtained in [KY], when the initial data be- long to certain BM spaces of negative index s. Similar results can be found in [Kr] concerning the spaces BEα. Our aim here is to give a unifying approach using the theory developed in the previous sections and extend the analysis to compact manifolds. Section 5.1 deals with background material and briefly introduces techniques which are used later. As discussed above, we look for mild solutions by applying the contraction mapping theorem or by direct inspection of the Picard iteration. In Section 5.2 we consider scalar perturbations of the heat equation, while section 5.3 treats the Navier-Stokes system. A typical feature of this approach is the existence of a critical index for the space of initial data—it is s = n/p 1, 1 p , for NS—which corresponds to so-called limit spaces, characterized− by invariance≤ ≤∞ of the norm under the natural scaling of the equation. In the subcritical case, i.e., s>n/p 1, one expects (local) existence for all initial data, while an extra condition needs− to be imposed in the critical case, but global existence then follows by means of time dilations. Under the identification (1.13), the results in [KY] are indeed covered by the existence theorem in [Kr] for the spaces BEα, and the condition on the initial data in [KY] is, in fact, redundant for subcritical indices. Additionally, we show persistence of regularity for spaces with positive index s, where condition (1.4) s holds. Specifically, Np,q,r is adapted to NS for (1) s>n/2p,ifp>n,1 q<2, r [1, ], (2) s>0, if p>n,2 q≤ p< , r∈ [1∞, ]. ≤ ≤ ∞ ∈ ∞ In Section 5.5, we consider the same equations on compact manifolds. In partic- ular, we prove that, provided the Littlewood-Paley operators ψj(D) are substituted by ψj(√ ∆), virtually all previous results extend. Because of the finite size of the domain,− the behavior for t 0 is the relevant question in this case. The analy- sis is slightly more complicated∼ for NS than for scalar semi-linear equations, since curvature adds an extra term to the equation. Summarizing, short-time existence and uniqueness for Navier-Stokes holds in s Np,q, ,if ∞

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n/p 1 n/2p, p>n, for arbitrary initial data, • s = n/p− 1 for data satisfying an extra condition. • − n/p 1 The solution persists for all times for sufficiently small initial data in p,q, − . We conclude by introducing some notation that will be used throughoutN ∞ the paper. 1 First of all, and − denote respectively the Fourier and inverse Fourier transforms, andF we set F

1 ( f)(ξ)=fˆ(ξ), ( − g)(x)=ˇg(x). F F is the Schwartz class of rapidly decreasing functions and and its dual 0 is the spaceS of tempered distributions. We indicate continuous injections with , S. → Unless otherwise noted, we consider functions or distributions on Rn,where n 2 is fixed, and we usually omit the reference to the underlying space, i.e., we ≥ p p p n p n write L , q for L (R ), q(R ), and so on. Mt∆ M With e , t 0, we mean the Weierstrass or heat semigroup and pt is the corresponding Schwartz≥ kernel. Recall that 1 x y 2 pt(x)= exp −| − | . (4πt)n/2 { 4t }

D stands for the differential operator (1/i) ,wherei = √ 1, so that (Djf)(ξ) ∇ − F = ξj fˆ.IfP is a pseudo-differential operator, σ(P ) will denote its symbol and σ0(P ) its principal symbol. Finally, we denote any irrelevant constant with C, unless we want to emphasize that C depends on certain indices, say p or q,inwhichcasewewriteCp,q or C(p, q).

2. Two Classes of Function Spaces It is a common experience that different time and length scales appear in the motion of fluids and that certain phenomena seem to be independent of the choice of such scales. Mathematically, scaling properties of non-linear differential equations are an important tool in analyzing solutions, especially when the non-linearity is a homogeneous function. Invariance under some scaling law allows us, for example, to construct new so- lutions from known ones. Also, self-similar solutions, if they exist, persist for all times. For the Navier-Stokes system, global-in-time existence of smooth solutions has not been proven to this day, except for special cases. It is, therefore, natural to choose initial data in function spaces that are con- structed by imposing some scaling behavior. This behavior can be tailored to the type of non-linearity in the equation.

2.1. Besov-Morrey spaces. Due to the smoothing effect of the heat semigroup, solutions to the Navier-Stokes equation exist when the initial data have minimal regularity—for example, they can be distributions whose associated vorticity is a singular measure (see [GM], [GMO]). This motivates in part the study of the Navier-Stokes and other evolution equations in Morrey spaces and generalizations. Examples of such spaces were constructed by H. Kozono and M. Yamazaki ([KY]). We start by recalling the definition and some of the properties of the Morrey spaces p and M p. Mq q

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Definition 2.1. For p and q satisfying 1 q p< , define the homogeneous p ≤ ≤ ∞ p Morrey space, q,andtheinhomogeneous or local Morrey space, Mq , respectively as M p q n/p n/q p q (2.1) q f Lloc f q =supsup R − f L (B(x0,R)) < n M ≡{ ∈ |k kM x0 0

p p0 M M if 1 q p0 p< , q ⊂ q ≤ ≤ ≤ ∞ (2.4) M p M p and p p if 1 q2orq = p = 2. On the other hand, L∞ Mq for all values of p and ∈q. ⊂ Morrey spaces can be seen as a complement to Lp spaces. As a matter of fact, p p p q p L and q L . They are part of a larger class, that of Morrey-Campanato M ≡ M ⊂ p spaces q,λ, which also include Lipschitz spaces and BMO. Mq corresponds to the choice ofL parameter λ = n(1 q/p)(see[Pe]). Morrey spaces describe local− regularity more precisely than Lp. For example, Morrey’s Lemma gives p n/p n (2.5) M1 C− (R ) . ⊂ ∗ p Note that it is stronger than the usual Sobolev embedding, since M1 is strictly larger than Lp. (2.5) is a consequence of the characterization below in terms of heat kernel estimates [Tay3]. If f L1 ,then ∈ loc p n/2p (2.6) f M pr f Cr− , ∈ 1 ←→ ∗| |≤ for 0 1. These are sharp results. Consequently, the heat semigroup is not strongly continuous on Morrey spaces. As anticipated in the Introduction, some of the interest for Morrey spaces in fluid mechanics is related to spaces of measures. Most of the properties carry over if we replace the integral of f in Definition 2.1 with the total variation of a Radon ˜ p p measure µ. The corresponding space is sometimes indicated by [Tay3] and 1 corresponds to the subspace of measures absolutely continuousM with respect toM the Lebesgue measure. Following [KY], we now define Besov spaces based on M p and p. q Mq

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We first recall the concept of Littlewood-Paley partition of unity, ψj , j { } ∈ Z. The functions ψj(ξ) are constructed as follows. Let ϕ0 be a smooth function supported in the ball B(0, 1), such that ϕ0 1for0 ξ 2/3. Set ϕj (ξ)= j ≡ j ≤||≤ ϕ0(ξ/2 ), so that ϕj is supported on the ball B(0, 2 ). Define ψj = ϕj+1 ϕj.Note j −n j 1 that ψj(ξ)=ψ0(ξ/2 ). ψj is supported on the dyadic shell Dj = ξ R 2 − < j+1 { ∈ | ξ < 2 . The collection of all shells Dj,asj runs over the integers, covers | n| } Rξ 0 ,withD` Dk = only if k ` > 1. Although the width of each shell \{ } ∩ ∅ | − | j doubles in size as j increases, Dj is still centered around the frequency ξ =2 .In | | fact, the ψj’s act as band filters in the language of digital processing, while the φj are low pass filters. If f 0,then ∈S (2.8) f =(ϕ0 + ψj)f, j 0 X≥ with convergence in the sense of distributions (telescopic sum). On the other hand, in general, ( ψj)f = f,sinceψju =0in for every distribution u that is j Z 0 supported at the∈ origin.6 S P 3 Finally, let ψj(D) be the Fourier multiplier with symbol ψj defined as

1 ix ξ (2.9) ψ(D)u(x)= − (ψj uˆ)(x)= ψj(ξ)ˆu(ξ) e · dξ , u . F · n ∈S ZR Definition 2.2. For 1 q p< , r [1, ],ands R, we say that f 0 s ≤ ≤ ∞ ∈ ∞ ∈ ∈S belongs to the space Np,q,r if

1/r sj r (2.10) f N s = ϕ0f p + 2 ψj(D)f p < , k k p,q,r k kMq  k kMq  ∞ j 0 X≥
 and, similarly, we say that the equivalence class of distributions modulo polynomials, s f 0/ , belongs to the space if ∈S P Np,q,r 1/r sj r s p f = 2 ψj(D)f q (2.11) k kNp,q,r  k kM  j X∈Z
 sj p = 2 ψj(D)f q j∞= r < .  { k kM } −∞ ` ∞ The spaces s and N s are hence defined respectively as the homogeneous p,q,r p,q,r N ˙ s s p p p and inhomogeneous Besov space Bp,r, Bp,r with q and Mq replacing L .For s s M brevity, we will call p,q,r and Np,q,r respectively the homogeneous BM space and inhomogeneous BM spaceN . It is not difficult to see that different choices for the family of functions ψj { } give equivalent norms, as long as the ψj form a Littlewood-Paley partition of unity.

s Remark 2.3. One could consider constructing a non-local version of Np,q,r by tak- js p p r ing distributions such that 2 ψj(D)f Mq j Z ` < . However, Mq is not invariant under dilations, andk{ so thek norm ofk the} ∈ correspondingk ∞ homogeneous space would not have a definite scaling degree.

3See also Section 3.1.

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s s As discussed in [KY], the spaces p,q, and Np,q, are strictly larger then Mor- N ∞ ∞ 1 rey spaces; for example, the principal value distribution p.v. x belongs to 1,1, , 1 N ∞ but not to 1. On the other hand, their behavior in frequency space is simpler and they areM a better candidate for the analysis of partial differential equations, as we aim to show in the paper. They are also strictly larger than the corresponding Besov spaces, when p>q. Let us, again, collect some useful inclusion relations. From (2.4) and the fact r r0 that ` ` if r r0, it follows immediately that ⊂ ≤ (2.12) N s N s and s s p,q,r ⊂ p,q0,r0 Np,q,r ⊂Np,q0,r0 r if 1 q0 q p< and 1 r r0 .H¨older’s inequality for ` -spaces gives instead≤ ≤ ≤ ∞ ≤ ≤ ≤∞ (2.13) N s N s0 , p,q,r ⊂ p,q,r0 for every r, r0 [1, ]ifs s0. Hence, the inhomogeneous spaces form a scale. ∈ ∞ ≥ s In particular, s can be interpreted as a “smoothness” index, i.e., elements of Np,q,r are more and more regular as s increases (cf. Theorem 2.4 below). However, we never have s s0 , intuitively because low and high frequencies “weigh the p,q,r p,q,r0 same” in theN Fourier⊂N representation of f s . ∈Np,q,r In general, a Littlewood-Paley component ψj(D)f may not decay at infinity, but it is slowly increasing by virtue of the Paley-Wiener theorem. The Morrey norm essentially controls the growth over balls of given radius. Therefore, by carefully decomposing the convolution ψj(D)f = ψˇj f, one can prove the following Sobolev- type embedding theorem. ∗ Theorem 2.4 ([KY]). The following inclusions hold: s s n/p s ˙ s n/p (2.14) Np,q,r B −,r , p,q,r B −,r , ⊂ ∞ N ⊂ ∞ if 1 q p< , s R and r [1, ]. ≤ ≤ ∞ ∈ ∈ ∞ In the critical case s = n/p; in particular, we have (we state only the homoge- neous case for simplicity) n/p ˙ 0 ˙ 0 p,q, B , C , N ∞ ⊂ ∞ ∞ ≡ ∗ n/p ˙ 0 (2.15) p,q,r B ,2 BMO,r [1, 2], N ⊂ ∞ ⊂ ∈ n/p L∞, Np,q,1 ⊂ where C˙ s are homogeneous Zygmund spaces [Trieb2] and BMO is the space of functions∗ of bounded mean oscillations, introduced by John and Nirenberg [JN]. s s k k Recall that C C if s>0 is not an integer, but C C for k Z+. ∗ ≡ n/2p p ⊂ ∗ ∈ Potential estimates give ( ∆)− q , BMO [Miy]. So, (2.15) follows im- mediately from a theorem of− Littlewood-PaleyM → type for Morrey spaces [Maz2]. From Theorem 2.4, completeness of BM spaces follows (cf. proof of Lemma s s 4 2.21). Also, the inclusions p,q,r , 0/ , Np,q,r , 0 are continuous. Ns →S P →S In general, elements of belong only to 0/ , since the kernel of each opera- Np,q,r S P tor ψj(D) contains those distributions for which the Fourier Transform is supported at the origin. However, for certain values of p, q and s,onecancanonically identify s each class f in p,q,r with a distribution. To do so, note that for each fixed l Z, N ∈ 4 / is endowed with the quotient topology. S0 P

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∞ ψj(D)f converges to an element, f ,of 0. Using the fact that the supports j=l (l) S of ψj(ξ)andψk(ξ) do not overlap for j k > 2, one can show that actually the sequenceP f converges in s and| to−f.| So, it is enough to prove that f { (l)} Np,q,r { (l)} converges in 0. This will depend on the values of p, q and s. S Proposition 2.5 ([KY]). Let s

Proposition 2.6 ([KY]). For s1 = s2 R and θ (0, 1), the following interpola- tion relations hold: 6 ∈ ∈

s1 s2 s (a)(Np,q,r1 ,Np,q,r2 )θ,r = Np,q,r and s1 s2 s ( p,q,r1 , p,q,r2 )θ,r = p,q,r , (2.16) N N N s1/2 p s2/2 p s (b)((I ∆)− M , (I ∆)− M )θ,r = N and − q − q p,q,r s1/2 p s2/2 p s (( ∆)− , ( ∆)− )θ,r = , − Mq − Mq Np,q,r where s =(1 θ)s1 + θs2, r, r1,r2 [1, ],and1 q p< . − ∈ ∞ ≤ ≤ ∞ The proof is essentially an adaptation of the Lp case. Other useful relations follow in a similar fashion. For the sake of brevity, we consider homogeneous spaces only.

Proposition 2.7. Again, let s, sj R, 1 q, qj p , r, rj [1, ], j =1, 2, θ (0, 1).Then ∈ ≤ ≤ ≤∞ ∈ ∞ ∈ 1 1 θ θ (1) ( s , s ) = s ,if = + , p,q,r1 p,q,r2 θ,r∗ p,q,r∗ − N N N r∗ r1 r2 1 1 θ θ (2) ( s1 , s2 ) s ,ifs =(1 θ)s + θs , = + , p,q1 ,r1 p,q2,r2 θ,q∗ p,q∗ ,r∗ 1 2 − N N ⊂N − r∗ r1 r2 1 1 θ θ = − + and q∗ = r∗. q∗ q1 q2 Proof. Following [BL] (pp. 149-153), it is enough to show that s is a retract Np,q,r of `˙s( p), where r Mq + s p p ∞ jsr r 1/r `˙ ( )= α ,α ,n ( 2 α p ) . r q n n q Z j q M { } ∈M ∈ | k kM ≤∞ j=  X−∞  ˙s p For α = αj ` ( ), define a map by  { }∈ r Mq I

(2.17) (α)= Ψj αj , I ∗ j X j+1

where Ψj = ψk;so,Ψj is identically one on the support of ψj. Similarly, for k=j 1 X− f 0, define a map by ∈S J (2.18) (f)j = ψj f. J ∗

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Then, clearly is the identity map on 0/ . To prove our claim, we need to show that I◦J S P : s `˙s( p) J Np,q,r → r Mq and : `˙s( p) s P r Mq →Np,q,r p s boundedly. But (f) `˙s( ) f p,q,r by definition. Vice versa, since Ψj ψk kJ k r q ≈k kN ∗ ≡ 0if k j > 1, M | − | 1/r ks r s p (α) C (2 ψk αk q ) (2.19) kP kNp,q,r ≤ k ∗ kM "k # X∈Z C α s p . `˙ ( q ) ≤ k k r M The last inequality follows from the following simple observation, which will be frequently exploited. Remark 2.8. If E is any normed vector space of distributions, whose norm is translation-invariant, then the Fourier multiplier φ(D) is bounded on E for all φ such that φˇ L1 and ∈ (2.20) φ(D) op φˇ 1 . k k ≤k kL In particular, the Littlewood-Paley operators are uniformly bounded with

ψj(D) op 1, j. k k ≤ ∀ ˙s To conclude the proof, we use the following results on interpolation of `r(A) spaces, where A is any Banach space ([BL], pp. 121-124):

(2.21) `˙s (A), `˙s (A) = `˙s (A) , r1 r2 r∗ θ,r∗   which proves part (1), and

(2.22) `˙s1 (A ), `˙s2 (A ) = `˙s ((A ,A ) ) r1 1 r2 2 r∗ 1 2 θ,q∗ θ,r∗   if r = q . Part 2 now follows from the inclusion ( p , p ) p ,whichis ∗ ∗ q1 q2 θ,q∗ q∗ qM M ⊂M in turn a simple consequence of interpolation of L spaces.  Contrary to the case of Besov spaces, the inclusion in part 2 is strict. The op- posite inclusion does not hold, because Morrey spaces are not interpolation spaces. As shown in [BRV], it is possible to construct operators T bounded from p into Mqj qj p q∗ L that are not bounded from q into L . The above proposition will beM useful∗ in Section 3, when proving regularity of pseudo-differential operators in the spaces s and N s . Np,q,r p,q,r 2.2. Wavelet decomposition. In this section, we are going to develop a decom- s s position of elements of p,q,r and Np,q,r in terms of quasi-orthogonal “smooth wavelets”, similar to thatN obtained by Frazier and Jawerth for Besov spaces [FJ1]. The type of wavelets used here are not compactly supported, although they are con- centrated on compact sets (dyadic cubes); on the other hand, they have infinitely many vanishing moments. Wavelets have been used in the analysis of the Navier-Stokes equation [Fe], yet there does not seem to be a clear advantage over the Littlewood-Paley approach

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(cf. [Can]). On the other hand, wavelets have been successful in treating the Euler equation, where there is no smoothing effect of the heat kernel [Vi2]. n Let us begin defining dyadic cubes Qνk,forν Z and k Z : ∈ ∈ n ν ν (2.23) Qνk = x R 2− ki xi 2− (ki +1),i=1,...,n . { ∈ | ≤ ≤ } n For fixed ν, the collection Qνk,k Z tiles the whole space and the cubes are { ∈ } ν pairwise disjoint. Each cube Q = Qνk is uniquely identified by its length `(Q)=2− ν andapreferredcornerxQ =2− k.WealsodenotethevolumeofthecubeQ with Q . | |The smooth wavelets we are going to employ arise naturally from Littlewood- Paley components. The first step is contained in the following lemma (see [FJ1]).

Lemma 2.9. Let f 0/ and let ψν be a (homogeneous) Littlewood-Paley partition of unity. Then∈S P { }

nν ν ν (2.24) f( )= 2− ψˇν f(2− k) ψˇν ( 2− k) , · ∗ ·− ν k n X∈Z X∈Z ν where the convergence is in 0/ and, as before, ψν (ξ)=ψ(ξ/2 ). S P The main idea behind the proof is to exploit the compact support of ψν to extend ˆ ν+1 n ψν f to a periodic function of period 2 π in Rξ , and then use Fourier series to ν represent it as a discrete sum of “wavenumbers” 2− k coupled to the frequency ξ. f is now a superposition of elements of a given family of test functions with ˇ ν coefficients equal to the value of fν = ψν f at the sampling points 2− k. ∗ n For notational convenience, as we will be working in physical space Rx,weset νn ν ψˇν (x)=σν (x)=2 σ(2 x) ,σ= ψˇ0 , and ν σνk(x)=σ(2 x k) . − Then, after some simple manipulations, (2.24) becomes

νn (2.25) f = 2− (σνk,f) σνk , ν k X X which is an almost orthogonal decomposition, since (σνk,σµl)=0if ν µ > 1, | − | or anyway negligible if k l is large enough. Each σνk is a smooth wavelet | − | concentrated on the dyadic cube Qνk with infinite vanishing moments, i.e.,

M (2.26a) x σνk(x) dx =0, Rn Z ν ν M (2.26b) σνk(x) CM (1 + 2 (x 2− k))− | |≤ − for all M 0. ≥ The next step is to rewrite the norm of each Littlewood-Paley component fν ap- propriately in terms of dyadic cubes. For Lp spaces, a classical result of Plancherel- Polya holds [FJ1]. ˆ ν Lemma 2.10. Let 0

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We will prove a similar lemma in the case of Morrey spaces. Clearly, we do not expect to have a full sum over the wave-vectors k, but only a partial sum.

ˆ ν Lemma 2.11. Let 1 q p< , ν Z, f 0.SupposeSupp f B(0, 2 ). Then, ≤ ≤ ∞ ∈ ∈S ⊂

1/q 1 q/p 1 − 1 q/p q sup Q − sup f(x) ν (2.28) `(J) 2 J | | Qνk | |  −   Qνk J ≥ | | X⊂  ν(n/p) p  Cn,p,q 2 f q , ≤ k kM where J are dyadic cubes.

Proof. We can choose test functions σν satisfying all the hypotheses of Lemma 2.9 such thatσ ˆν 1 on the support of fˆ. Hence, from (2.24), ≡ νn ν ν f(x + y)=σν fy(x)=2− f(2− k + y) σν (x 2− k) ∗ − k n (2.29) X∈Z ν = f(2− k + y) σνk(x) , k n X∈Z

with fy(x)=f(x + y). But σνk is concentrated on the dyadic cube Qνk. Therefore (2.26b) easily gives with M = n +1,

sup f(z) sup f(x + y) z Qνk | |≤x Qν0 | | ∈ ∈ sup f(x + y) ≤ x <2 ν √n | | | | − ν Cn f(2− l + y)supσνl(x) ≤ ν | | l n x <2− √n X∈Z | | ν (n+1) Cn f(2− l + y)(1 + l )− , ≤ | | l n X∈Z

where y is some fixed point in Qνk. Using the triangle inequality (q =1)orH¨older’s inequality (q>1), and integrating both sides with respect to y gives

νn q (n+1) ν q 2− sup f(z) Cn,q (1 + l )− f(2− l + y) dy z Qνk | | ≤ | | Q | | ∈ l n νk (2.30) X∈Z Z (n+1) q Cn,q (1 + l )− f(y) dy . ≤ | | | | l n Qνk+l X∈Z Z ν ν Clearly, a dyadic cube J can be covered by cubes of length 2− only if `(J) 2− , so that ≥

νn q (n+1) sup 2− sup f(x) Cn,q (1 + l )− ν `(J) 2 x Qνk | | ≤ n | | − Qνk J ∈ l ! ≥ X⊂ X∈Z sup f(y) q dy . ν · `(J) 2 Qνk | | − Qνk J ≥ X⊂ Z

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In the last step we have trivially shifted all cubes along the dyadic grid, which does not affect the supremum. Finally, (2.31) νnp/q n(q/p 1) 1 q/p q sup 2− `(J) − Q − sup f(x) ν `(J) 2 | | x Qνk | | − Qνk J ≥ X⊂ ∈ n(q/p 1) νn q sup `(J) − 2− sup f(x) ν ≤ `(J) 2 x Qνk | | Qνk J ≥ X⊂ ∈ n(q/p 1) q Cn,q sup `(J) − f(y) dy ≤ `(J) 2ν J | | ≥ Z q Cn,q f p . q ≤ k kM 

If f is any distribution, regardless of its support, replacing f with fν in the above lemma easily gives (2.32) r 1/r 1 q/p νn(s/n 1/p) 1 − νn q 1/q 2 − (sup 2− sup fν (x) ) ν   `(J) 2 J x Qνk | |   ν Z −   Qνk J ∈ X∈ ≥ | | X⊂   1/r   sν r p s C 2 σν f q = C f , ≤ k ∗ kM k kNp,q,r "ν # X∈Z
for all s R,1 q p< , r [1, ]. Following∈ [FJ1],≤ let≤ us introduce∞ ∈ the∞ following suggestive notation.

Notation 2.12. If Q = Qνk,thenset

s/n 1/p σQ = Q − σνk , | | ν(s n/p) ν sQ =2 − σν f(2− k) . ∗ We can collect all previous results to obtain the following wavelet representation for the space s . Np,q,r Proposition 2.13. Let f s .Then ∈Np,q,r

f = sQ σQ Q dyadic X in 0/ ,and (2.33)S P r/q 1/r 1 q/p 1 − 1 q/p q C(n, p, q, r) f s sup Q − sQ . p,q,r   k kN ≥ `(J) 2 ν J | | | |  ν Z ≥ − | |  Qνk J X∈ X⊂      Note there is no explicit dependence on the index p and s in this decomposition. It is, in fact, hidden in the definition of σQ and sQ.Also,forp = q, we correctly recover the decomposition for Besov spaces (see [FJ1]).

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Proof. The representation formula for f follows directly from Lemma 2.9. As for

(2.33), it is enough to notice that Q sQ σQ with Q = Qνk satisfies the hypotheses of Lemma 2.11 for ν fixed. Hence, (2.32) becomes P 1 q/p νn(s/n 1/p)q 1 − s C f p,q,r 2− − sup k kN ≥ ν J · "ν `(J) 2−   X∈Z ≥ | | r/q 1/r 1 q/p q Q − sup sQ σQ(x)  · | | x Qνk | |  Qνk J Qνl X⊂ ∈ X    r/q 1/r 1 q/p 1 − 1 q/p q = sup Q − sup sQ σνk(x) .  ν  `(J) 2 J | | x Qνk | |  ν Z ≥ − | | Qνk J ∈ Qνl X∈ X⊂ X      But the cubes Qνk are pairwise disjoint. So (2.26) easily implies that the largest contribution to the sum above is attained when k = l.  It is straightforward to prove analogous propositions for the inhomogeneous s spaces Np,q,r. First of all, replace the homogeneous Littlewood-Paley partition of unity ψν , ν Z, with the corresponding inhomogeneous one φ0,ψν , ν 0 { } ∈ { } ≥ (using the notation introduced in Section 2.1), and let sk =ˇϕ0 f(k). Second, since p ∗ we take the local Morrey space Mq as base space, we must restrict dyadic cubes to those of length `(Q) 1. ≤ Proposition 2.14. Let f N s .Then ∈ p,q,r f = sk ϕˇ0( k)+ sQ σQ ·− νk νk k n ν 0 k X∈Z X≥ X∈Z in 0,and S (2.34)

s C f Np,q,r sup sk k k ≥ k n | | ∈Z r/q 1/r 1 q/p 1 − 1 q/p q +  sup Q − sQ  . 2 ν `(J) 1 J | | | |  ν 0 −   Qνk J X≥ ≤ ≤ | | X⊂     Remark 2.15. The reverse of inequality (2.33) (and similarly (2.34)) can be shown

to hold. If sQ is any sequence indexed by dyadic cubes, f = Q sQσQ in the sense of distributions,{ } and P r/q 1/r 1 q/p 1 − 1 q/p q (2.35) f =  sup Q − sQ  < , k k∗ `(J) 2 ν J | | | |  ∞ ν Z ≥ − | | Qνk J X∈ X⊂  s     then f p,q,r with f N s f . More generally, σQ can be replaced by an ∈N k k p,q,r ∼k k∗ (s, p)-molecule, as is the case for Besov spaces [FJ1]. We do not have to decompose molecules concentrated on dyadic cubes in atoms supported on cubes (cf. [Uch] and the decomposition for BMO), because cancellations are not crucial in this context. Details are provided in [Maz2].

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The above construction should also be compared with a similar one for the space BMO, which follows from work of Uchiyama [Uch]. Using the above notation, we have 1/2 + ∞ 1 νn 2 (2.36) f BMO sup 2− sup σν f , k k ≈ J dyadic  J Qνk | ∗ |  ν= log (`(J)) Qνk J − X2 | | X⊂   n/p and it is then easy to see, for example, that p,q,r BMO for q 2, r 2. Frazier and Javerth have extended Uchiyama’sN ⊂ result to the≥ whole≤ class of ˙ s Triebel-Lizorkin spaces F ,r,1 r< , s R [FJ2]. Their decomposition ∞ ≤ ∞ ∈ n/p 1 along with (2.33) allows us to compare the Besov-Morrey space p,q, − , p>n, 1 N ∞ with BMO− . We are particularly interested in the relation between these spaces in connection with well-posedness at the critical scaling for the Navier-Stokes equa- tion. We refer to Section 5 for a more extensive discussion. 1 The space BMO− is the space of tempered distributions that can be written 1 as divergence of a vector field with components in BMO. The norm in BMO− is given by

1/2 R2 n t∆ 2 (2.37) f 1 sup sup R e f(y) dt dy . BMO− − k k ≈ x n R>0 B(x,R) 0 | | ∈R Z Z ! Using the characterization (2.57), which is proved in the Section 2.3, it is easy to see that n/p 1 1 p,q, − BMO− ,q 2,p>n. N ∞ ⊂ ≥ However, for q =1,

n/p 1 1 (2.38) p,1, − * BMO− N ∞ 1 n/p 1 (and similarly BMO− * p,1, − ). N ∞ 2.3. Scaling properties: the spaces BEα. Recently, G. Karch introduced a large class of Banach spaces based on scaling properties of the heat semigroup, and used them to study certain semi-linear parabolic equations [Kr]. BM spaces belong to this class for a particular range of indices. We begin again with some definitions and background notation.

Definition 2.16. Let (E, E) be a functional Banach space, that is, , E, k·k S → → 0.Setfλ(x)=f(λx), λ>0.Thenorm E has scaling degree equal to κ,if S ∀ κk·k f E and λ>0, fλ E and fλ E = λ f E. ∀ ∈ ∀ ∈ k k k k p We will denote a space with scaling degree n/p by Ep. Both the space L p − and the (homogeneous) Morrey space q have scaling degree equal to n/p.BM spaces have the same scaling degree asM Besov spaces (namely s n/p), while− being strictly larger. This property will be exploited in studying differential− equations. We implicitly assume that E always has some scaling degree, which is true for all classical spaces. Norm estimates for the heat semigroup can be easily obtained by rescaling ar- guments.

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t ∆ Lemma 2.17 ([Kr]). Suppose e 0 : Eq Ep is a bounded linear operator for → some t0 > 0.Then,thereexistsC = C(p, q, t0) > 0 such that t∆ n(1/p 1/q)/2 (2.39) e f E Ct − f E , t>0 , f Eq . k k p ≤ k k q ∀ ∀ ∈ Moreover, if q>p, then necessarily Eq = 0 . { } α/2 t∆ In view of the above Lemma, one expects t e f Ep to be bounded for some α>0, even if f belongs to a weak space of distributions.k k Definition 2.18. Given a functional Banach space E, define α α/2 t∆ (2.40) BE = f 0 f BEα sup t e f E < . { ∈S |k k ≡ t>0{ k k } ∞} It is not difficult to see that BEα = 0 for α<0, while E BE0, although this space is not as interesting. Therefore,{ we} will implicitly assume⊂ in the following that α>0. α Lemma (2.17) implies that Eq BEp q such that n(1/q 1/p)/2 α/2if t∆ ⊂ ∀ − ≤ e : Eq Ep is bounded for some t>0. In particular, → (2.41) Lq BEα for E = Lp if 1 q p and α = n(1/q 1/p) , ⊂ ≤ ≤ ≤∞ − p1 α p2 (2.42) BE for E = if 1 p1 p2 and α = n(1/p1 1/p2) . Mq1 ⊂ Mq2 ≤ ≤ ≤∞ − In the case E = Lp,thespaceBEα is, in fact, the homogeneous Besov Space ˙ α Bp,− . This is a consequence of the following characterization of Besov spaces in terms∞ of the Weierstrass semigroup (see [Trieb1], page 192):

Theorem 2.19. For every f 0, the following norms are equivalent: ∈S f α , B˙ p,− k k ∞ α/2 t∆ sup t e f Lq . t 0 { k k } ≥ Intuitively, Theorem 2.19 holds because the symbol of the operator et∆ and the symbol of ϕj (D) have a similar behavior in frequency. The spaces BEα are complete if the class of underlying E’s is appropriately restricted. Definition 2.20. denotes the class of functional Banach spaces E such that et∆ : E Lq is boundedK t>0 for some q [1, ]. → ∀ ∈ ∞ For example, the Morrey spaces p have this property with q = . Mq ∞ α Lemma 2.21 ([Kr]). Assume Ep .Then(BE , BEα ) is a Banach Space. ∈K p k·k p α ˙ γ In fact, if Ep , by Lemma 2.17, BEp , Bq, ,whereq p and γ = ∈K → ∞α ≥ ˙ γ α n(1/p 1/q). So, a Cauchy sequence fk in BEp converges to f in Bq, . − − −t∆ { } ∞ Moreover, e fk is a Cauchy sequence in Ep for t>0 and, therefore, converges { } α to some g(t)inEp.Sincethefk’s are uniformly bounded in BE , we can conclude α/2 that g(t) Ep grows in t at most as t− ,ast 0. So, it is enough to show that k t∆k → g(t)=e f in 0. But this is true by continuity of the heat semigroup in 0 and S S because Ep , 0. Note that,→S if E has scaling degree κ,thenBEα has scaling degree κ α.So, estimates similar to (2.39) hold for certain values of α. − In view of Theorem 2.19, we expect all BEα spaces to be characterized in terms of a Littlewood-Paley partition of unity.

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Proposition 2.22. Let E be a functional Banach space such that the norm has some scaling degree γ>0. Then, an equivalent norm in BEα is given by jα (2.43) f BE∗ α =sup 2− ψj(D)f E . k k j { k k } ∈Z For clarity, we split the proof into different lemmas.

α Lemma 2.23. If f BE ,thenψj(D)f E, j Z, and there is a constant C = C(E,α) such that∈ ∈ ∀ ∈ jα (2.44) sup 2− ψj(D)f E C f BEα . j { k k }≤ k k ∈Z j n j 1 Proof. Set t =4− .Sinceψj(ξ) is supported on the shell Dj = ξ R 2 − < ξ < 2j+1 , { ∈ | | | } 1 ψj(D)f E = − (ψj (ξ)fˆ(ξ)) E (2.45) k k kF k 1 t ξ 2 C˜ − (e− | | ψj (ξ)fˆ(ξ)) E , ≤ kF k so that jα ˜ α/2 ˇ 1 t ξ 2 ˆ 2− ψj(D)f E Ct ψj L1 − e− | | f(ξ) E (2.46) k k ≤ k k kF k α/2 1 t ξ 2 α/2 t∆ Ct − e− | | fˆ(ξ) E Ct e f E , ≤ kF k ≤ k k by means of Remark 2.8.  jα Lemma 2.24. If f 0 is such that supj 2− ψj(D)f E < ,then ∈S ∈Z{ k k } ∞ jα jα (2.47) sup 2− ϕj(D)f E sup 2− ψj(D)f E . j { k k }≈j { k k } ∈Z ∈Z k Proof. It is enough to show that, for each fixed k, ϕk(D)f = j= ψj(D)f in E. (k) k −∞ Let Pl = j= l ψj(D)f.Setm l and consider − ≥ P P l 1 l 1 (k) (k) − − − − jα jα P P E C ψj(D)f E C 2 sup 2− ψj(D)f E , k m − l k ≤ k k ≤   { k k } j= m j= m j Z X− X− ∈ where the last term tends to 0 as l, m ifα>0. Hence, →∞ k kα kα 2− ϕk(D)f E 2− ψj(D)f E k k ≤ k k j= X−∞ k (j k)α jα 2 − sup 2− ψj(D)f E ≤   { k k } j= j Z X−∞ ∈  0  jα jα = 2 sup 2− ψj(D)f E   { k k } j= j Z X−∞ ∈  jα C sup 2− ψj(D)f E . ≤ j { k k } ∈Z The reverse inequality is obvious.  Note that this lemma would be clearly false if α were negative or zero.

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Lemma 2.25. Suppose that the norm of E has scaling degree equal to some γ>0. Then,

jα (2.48) f BEα C sup 2− ψj(D)f E . k k ≤ j { k k } ∈Z Proof. By Lemma 2.24, it is enough to show

jα (2.49) f BEα C sup 2− ϕj(D)f E . k k ≤ j { k k } ∈Z α/2 t∆ We need to estimate t e f E. Using homogeneity of the norm, one can restrict to the case t = 1. First,k the semigroupk property of et∆ allows us to reduce easily k to t =4− for k Z. Then, if we set fλ(x)=f(λx)forλ>0, we have ∈ t∆ ∆ (2.50) e f 1 (x)= e f 1 (x) , √t √t
y which follows simply by changing variablesy ˜ = in the convolution pt f. √t ∗ Similarly, (2.51) nj j ϕj(D)f 1 (x)=2 (ˇϕ0(2 ) f 1 )(x) √t · ∗ √t nj j k =2 ϕˇ0(2 (x y))f(2 y) dy n − ZR n(j k) j k k =2 − ϕˇ0(2 − (2 x y˜))f(˜y) dy˜ n − ZR n(j k) j k =2 − ϕˇ0(2 − (x/√t y˜))f(˜y) dy˜ = ϕ(j k)(D)f 1 (x) . n − − √t ZR k
So, replacing f by f 1 in (2.49) and setting t =4− gives √t

α/2 t∆ α/2 ∆ t e f 1 E = t (e f) 1 E k √t k k √t k α/2 γ/2 ∆ = t t− e f E k k k(γ α) ∆ =2 − e f E k k jα sup 2− ϕj(D)f 1 E ≤ j { k √t k } ∈Z jα =sup 2− (ϕ(j k)(D)f) 1 E j { k − √t k } ∈Z jα kγ =sup 2− 2 ϕ(j k)(D)f E j { k − k } ∈Z (j k)α k(γ α) =sup 2− − 2 − ϕ(j k)(D)f E , j { k − k } ∈Z because E has scaling degree γ. By renaming indices j k j, it remains to show that − → ∆ jα (2.52) e f E C sup 2− ϕj(D)f E . k k ≤ j { k k } ∈Z Now, on the left-hand side apply a Littlewood-Paley partition to f,sothatitis more convenient to replace the supremum by a sum over j on the right-hand side.

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To do so, fix β and  0 such that β >α. Then, ≥ −

jβ j kα (2.53) 2− ϕj(D)f E 2 2− sup 2− ϕk(D)f E . k k ≤   { k k } j 0 j 0 k Z X≥ X≥ ∈   Next, note that in 0, S j jβ jβ 2− ϕj(D)=cϕ0(D)+ 2− ψk(D) j 0 j 0 k=0 (2.54) X≥ X≥ X ∞ ∞ jβ = cϕ0(D)+ ψk(D) 2− . k=0 j=k X X But if Φk is a smooth compactly-supported function, which is identically one on ξ 2 the shell Dk, then the operator norm of the multiplier e−| | Φk(ξ) does not exceed kβ Cβ2− for every β 0, where Cβ does not depend on k (cf. Proposition 3.3). Hence, (2.54) implies≥ ∆ ∆ ∆ e f E e ϕ0(D)f E + e ψj(D)f E k k ≤k k k k j 0 (2.55) X≥ jβ Cβ 2− ϕj(D)f E , β. ≤ k k ∀ j 0 X≥  Equivalence of norms such as (2.43) are very well-known in the theory of function spaces, where they are a consequence of interpolation relations between semigroups of operators and their infinitesimal generators [BL], [Trieb2]. However, such results require that the semigroup be strongly continuous or even analytic. Our proof, in- stead, is based solely on some simple scaling arguments and requires the semigroup only to be uniformly bounded. Note, in particular, that the heat semigroup is not strongly continuous on Morrey spaces. α Consider now the homogeneous BM space p,q,− , α>0. By virtue of Proposi- tion 2.5, it can be canonically identified withN the set∞ of distributions jα f 0 sup 2− ∆jf E < . { ∈S | j { k k } ∞} ∈Z Therefore, Proposition 2.22 yields immediately5 α α (2.56) p,q,− = BE , N ∞ p p for E = q.Notethat q has scaling degree n/p; so Proposition 2.22 does indeed apply.M In turn, thisM identification gives an− equivalent norm in the space s p,q, , s<0, which is useful when dealing with semi-linear parabolic equations: N ∞ s/2 t∆ s p (2.57) f p,q, sup t− e f q . k kN ∞ ≈ t>0{ k kM } s Similar results hold for the local space Np,q,r. Proposition 2.26. Let 1 q p< , s<0.Then ≤ ≤ ∞ s/2 t∆ s p (2.58) f Np,q, sup t− e f Mq . k k ∞ ≈ 0

5This identification gives, in fact, a positive answer to a question of G. Karch [Kr].

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The proof is actually simpler, since no low-frequency part has to be considered.

s s Remark 2.27. Using the inclusion Np,q,r Np,q, for 1 r< , we immediately conclude that, if s<0, ⊂ ∞ ≤ ∞

s s/2 t∆ p (2.59) f Np,q,r sup t− e f Mq < . ∈ ⇒ 0

s/2 (2.60) f 0 Ct f s . Np,q, Np,q,r k k ∞ ≤ k k

3. Pseudo-Differential Calculus Our focus in this section is to study pseudo-differential operators in the function spaces previously introduced. Pseudo-differential calculus is a well-established tool for the analysis of partial differential equations, especially non-linear ones. Our results improve decisively upon those in [Kr] and [KY]. In particular, the theory developed here will be used extensively to define spaces on manifolds in Section 4. We will be mostly concerned with inhomogeneous spaces. Little can be said for α s the spaces BE ( p,q,r in particular), unless the symbol definition is appropriately modified. Intuitively,N the reason is the essentially local (or pseudo-local) nature of the operators we study. Their kernel can be highly singular along the diagonal, but it is smooth and rapidly decreasing away from it.

3.1. Basic estimates. For the sake of clarity, let us recall some basic definitions. A rather complete, but concise, treatment of the theory of pseudo-differential op- erators is given in [Tay1]. A classic, thorough reference is [Tay4]. Given a function p(x, ξ)onRn Rn, one can define an operator P ,actingsay on (if p does not grow too fast at× infinity), by S ix ξ (3.1) Pf(x)= p(x, ξ)fˆ(ξ) e · dξ . n ZR The function p is called the symbol, σ(P ), of the operator P . Fourier multipliers correspond to taking functions of ξ alone. We will also use the standard notation p(x, D) for the operator P with the function p as symbol.

Definition 3.1. Asmoothfunctionp(x, ξ) belongs to the symbol class Sm , m R, ρ,δ ∈ ρ, δ [0, 1],ifitsatisfies ∈ α β m α ρ+ β δ (3.2) D D p(x, ξ) Cα,β ξ −| | | | | ξ x |≤ h i for all multi-indices α and β. A pseudo-differential operator P belongs to the class m m OPSρ,δ if its symbol is a function in Sρ,δ. In (3.2), ξ stands for (1 + ξ 2)1/2.Thenumberm is the order of the operator, while (ρ, δ)isitsh i type. | |

m α1 αn Example 3.2. Let P = aα(x) ∂ ∂ be any m-th order differential α =0 1 ··· n m | | m α1 αn operator. Then, P OPS1,0 and σ(P )= α =0 aα(x) ξ1 ξn . ∈ P | | ··· P

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m The symbol class Sρ,δ was first introduced by H¨ormander [Ho]. Operators in this s s m m class behave nicely on Sobolev spaces, namely P : H H − for P S with → ∈ ρ,δ 0 δ<ρ 1. Also, as long as δ<1, P : 0 0. ≤Under the≤ above hypotheses, compositionS →S and taking adjoints are well-defined operations; precisely, m m (1) P OPS ,0 δ<ρ 1 P ∗ OPS , ∈ ρ,δ ≤ ≤ ⇒ ∈ ρ,δ

mj m (2) Pj OPS , j =1, 2 P1 P2 OPS with ρ =min(ρ1,ρ2)and ∈ ρj ,δj ⇒ ◦ ∈ ρ,δ δ =max(δ1,δ2)if0 δ2 <ρ 1. ≤ ≤ Finally, it is customary to set

m OPS−∞ = OPSρ, δ [0,1] . ∈ m R \ρ,δ∈

An operator P OPS−∞ is called a smoothing operator, because ∈ P : 0 C∞ , and (3.3) S → P : 0 . E →S Many other definitions and generalizations of symbol classes are clearly possible, and we will consider some later in the section. For the applications we have in mind, it is enough to take ρ = 1. On the other hand, δ will range over the whole interval [0, 1]. We now turn to establishing mapping properties of pseudo-differential opera- tors for BM and BEα spaces. Since the symbol of the Stokes operator (on Rn) has constant coefficients, the results found in [KY] and [Kr] concern the behav- ior of Fourier multipliers. Let ψj be the usual (homogeneous) Littlewood-Paley { } partition of unity subordinated to the dyadic decomposition of Rn into the shells Dj.

Proposition 3.3 ([KY]). Let m R and j Z.IfP (ξ) is a C∞ function on ∈ ∈ Dj 1 Dj Dj+1 = D˜ j such that P satisfies − ∪ ∪ α α (m α )j (3.4) ∂| |P/∂ξ A 2 −| | , | |≤ ˜ n for ξ Dj and some constant A, α N such that α [n/2]+1, then there exists ∈ ∀ ∈ p | |≤ p C>0 with the following property: u Mq (respectively q), 1 q p< , ∀ ∈ p M p ≤ ≤ ∞ such that Supp u Dj, we have P (D)u M (respectively )and F ⊂ ∈ q Mq mj P (D)u p CA2 u p k kMq ≤ k kMq (with a similar estimate for p). Mq It is enough to establish the following bound on the Schwartz kernel of P (D)Φ(D), K(x y): − mj (3.5) K 1 CA2 , k kL ≤ which is an easy consequence of (3.4) and the Plancherel theorem. Here, Φ is a ˜ function in C0∞ that is identically one on Dj. Proposition 3.3 readily gives continuity of Fourier multipliers in BM spaces.

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Proposition 3.4 ([KY]). Let m, s R, 1 q p< , r [1, ].Let N [n/2] + 1. ∈ ≤ ≤ ∞ ∈ ∞ ≥ a) If P (ξ) is a CN function on Rn 0 such that \{ } α α (m α ) (3.6) ∂| |P/∂ξ A ξ −| | , α N, | |≤ | | ∀| |≤ then s s m P (D): − Np,q,r →Np,q,r is a bounded linear operator. b) If P (ξ) is a CN function on Rn such that α α (m α ) (3.7) ∂| |P/∂ξ A ξ −| | , α N, | |≤ h i ∀| |≤ then s s m P (D): N N − p,q,r → p,q,r is a bounded linear operator.

m Note that if P (ξ) S1,0, then it satisfies (3.7) for every N. Similarly, following ∈ m n [Tay3], we define the symbol class Σ1 to be the set of smooth functions on R 0 m m \{ } such that (3.6) holds for every N.NotethatS1,0 Σ1 if m 0, but if m>0, condition (3.6) is stronger than (3.7), at least when⊂α

α α (m α )j (3.8) ∂| |P/∂ξ A 2 −| | , | |≤ ˜ n for ξ Dj and some constant A, α N such that α [n/2] + 1, then there ∈ ∀ ∈ | |≤ exists C>0 with the following property: u E such that Supp u Dj, we have P (D)u E and ∀ ∈ F ⊂ ∈ mj (3.9) P (D)u E CA2 u E . k k ≤ k k Proposition 3.7. Let m R.Supposes<0 and s

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m s Proof. Recall that an equivalent norm in BE − ,ifm s>0, is given by − j(m s) u m s sup 2 ψ (D)u . BE − − − j E k k ≈ j { k k } ∈Z It is therefore enough to prove that mj (3.11) ψj(D)P (D)u E C 2 ψj(D)u E , k k ≤ k k where P (D)andψj(D) commute, as scalar Fourier multipliers. Hence, we apply Lemma 3.6 with u replaced by uj = ψj(D)u and P (D)byP (D)Φ(D). On the shell j Dj, ξ 2 ,sothatP (ξ)Φ(ξ) satisfies (3.8). | |∼  The spaces BEα do not form a scale. Consequently, operators of negative index do not have the usual smoothing effect. In the following sections we will concentrate on inhomogeneous BM spaces, which do form a scale. Remark 3.8. The identification given by Proposition 2.22 breaks down if s>0, s since then BE− 0 , while the spaces defined as ≡{ } sj f 0/ sup 2 ψj(D)f E < ,sR , { ∈S P| j { k k } ∞} ∈ ∈Z are non-trivial even if s>0 (take E = Lp, which gives the classical homogeneous Besov spaces). The proof above still holds for these spaces when s>m, but they are not spaces of distributions in general. 3.2. Further results: non-regular symbols. Now we consider general pseudo- m differential operators p(x, D). The symbol class S1,δ is a little bit too restrictive for applications to non-linear equations, where typically differential operators have coefficients that depend on the solution itself. Therefore, while their symbol is still a smooth function of ξ, in general it will have only limited regularity in x. We will concern ourselves with symbols for which x-regularity is measured in H¨older or Zygmund spaces. Other scales of spaces can be used [Tay2]. Our choice is motivated especially by Theorem 2.4 in connection with para-differential calculus (cf. Section 3.3).

n n ` m Definition 3.9. The function p(x, ξ) on R R belongs to the symbol class C S1,δ, × ∗ δ [0, 1], `>0,ifitissmoothinξ and satisfies the following estimates: ∈ α α (m α +`δ) ∂ p( ,ξ)/∂ξ C` Aα ξ −| | and (3.12) k · k ∗ ≤ h i α α (m α ) ∂ p(x, ξ)/∂ξ Cα ξ −| | . | |≤ h i Similarly, one defines the class C`Sm , δ [0, 1], `>0, as the collection of symbols 1,δ ∈ p(x, ξ) such that α α (m α +`δ) ∂ p( ,ξ)/∂ξ C` Aα ξ −| | and (3.13) k · k ≤ h i α α (m α ) ∂ p(x, ξ)/∂ξ Cα ξ −| | . | |≤ h i The differences between the two classes are minor, especially if ` R+ Z. ∈ \ m Notation 3.10. If X is a function space, we will write p(x, D) XOPS1,δ if p(x, ξ) ` ∈ satisfies (3.12) with X in place of the C -norm. k·k ∗ Propositions 3.4 or 3.7 do not immediately extend to pseudo-differential oper- ators, in particular, because p(x, D)andψj(D) do not commute. Since ψj(D)is

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scalar, the commutator Q =[p(x, D),ψj (D)] will have lower order. However, it will not be a smoothing operator, even if p(x, ξ) is a regular symbol. m However, if p(x, ξ) S1,δ, δ<1, there is an asymptotic expansion for the symbol of Q: ∈ β i| | β β (3.14) q(x, ξ) D ψj(ξ) D p(x, ξ) , ∼− β! ξ x β >0 |X| and q(x, ξ) resembles a so-called elementary symbol (see, for example, [Bour2] or [CM]). m The asymptotic relation is given as follows. If q is a symbol in S1,δ,then ∼ m k q pj if, for each k>0, q pj S − .Inotherwords, pj(x, D) ∼ j 0 − j k ∈ 1,δ j converges≥ to q(x, D) modulo smoothings.≤ p is called the principal symbol of P P 0 P q(x, D), also denoted with σ0(q). It is customary to refer to j pj as “the” asymptotic expansion of q; although, clearly only the principal symbol is uniquely defined. P

If p j pj and each pj is a smooth function that is homogeneous of degree m j ∼for ξ large, then p is called a classical symbol of order m,andwewrite p −Sm. P ∈ cl ` m Definition 3.11. We call elementary symbol in the class C S1,δ, 0 δ 1,an expression of the form ∗ ≤ ≤

(3.15) σ(x, ξ)= σj (x)ψj (ξ) , j 0 X≥ j where ψ0 is smooth supported on the ball B(0, 2), ψj(ξ)=ψ(ξ/2 ) and ψ C0∞ n ∈ is supported on the dyadic shell D0 = ξ R 1/2 ξ 2 ,while σj is a uniformly bounded sequence such that { ∈ | ≤||≤ } { } j(m+`δ) (3.16) σj C` C 2 . k k ∗ ≤ ` m Convergence is intended in the Fr´echet topology of C S1,δ induced by the semi- norms ∗ α (`δ+m) α πN (σ)= sup ξ − ∂ξ σ( ,ξ) C` , n ξ R h i k · k ∗ α∈ N | |≤ α m α π˜N (σ)= sup ξ − ∂ξ σ( ,ξ) L∞ . n ξ R h i k · k α∈ N | |≤ 6 Example 3.12. If ψj is a Littlewood-Paley partition of unity and σj is a { } { } sequence uniformly bounded in C`(Rn),then ∗ jδ σ(x, ξ)= σj (2 x)ψj (ξ) j 0 X≥ ` 0 is an elementary symbol in C S1,δ. ∗ It is well-known ([Bour], [Ma]) that every symbol σ can be decomposed into a rapidly convergent sum of elementary symbols,

σ = σk , k n X∈Z 6 Here ψ0 corresponds to ϕ0 in our previous notation.

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where σk is such that πN (σk) CπN (σ). Informally, such a decomposition is obtained by first applying a Littlewood-Paley≤ partition of unity and then expand- 2 n ing each component in an orthonormal basis of L (Rξ ) to separate the x and ξ dependence. Therefore, the operator with symbol σ can be resolved into “elementary oper- ators” with symbols σk, which justifies restricting to operators with elementary symbols. Thus, whenever the norm of target and source spaces is defined itself in terms of Littlewood-Paley components, the decomposition of the symbol and the decomposition of the function can be “matched” and the problem is reduced to bound localized objects. This idea has been exploited to establish continuity of pseudo-differential opera- tors with non-regular symbols in inhomogeneous Sobolev spaces Hs,p and Zygmund spaces Cs [Ma], [Ru2], [Bour]. Here δ can be equal to 1; so the “exotic” H¨ormander 7 ∗ class L1,1 is included. However, the proof breaks down for the spaces Cs,whens 0. Indeed, a crucial step is given by the following lemma (see [Tay1], for∗ example).≤ ˆ j Lemma 3.13. Let f = j 0 fj in 0,with Supp fj B(0,A2 ) for some A>0. Then, for s>0, ≥ S ⊂ P js s (3.17) f C C(A)sup 2 fj L∞ . k k ∗ ≤ j 0{ k k } ≥ The restriction on the range of s cannot be lifted, since the hypotheses imply that the support of each Littlewood-Paley function ψj intersects the support of ks infinitely many fk.Buteach fk L∞ is weighted by a factor of 2− . So the total contribution is divergent for sk k0 (cf. proof of Lemma 3.16). For Sobolev spaces, duality and interpolation can≤ be used instead to extend the analysis to the case s<0. s Recall that the spaces BE− correspond to Besov-type spaces of negative index. So the above proof cannot be adapted in this context. On the other hand, even for the classical (homogeneous) Besov spaces, there is no analog to the lemma above if s is positive large enough. Clearly, there is no restriction on s if the fk’s are supported on dyadic shells instead of balls (cf. Lemma 3.16). But then, to use the same approach, one is forced m to consider symbols modeled on the class Σ1 . Due to the possible singularity of the symbol at the origin, convergence issues arise and there is no good calculus. For the homogeneous BM spaces, one could try and exploit the representation of s s p,q,r as real interpolation spaces (see Proposition 2.6). However, the spaces p,q,r N s/2 p N are actually better behaved than the spaces ( ∆) q . For example, the former are closed under the action of Fourier multipliers− ofM the Mikhlin-H¨ormander type while the latter are not (for q = 1) [Tay3], [Tay6]. Indeed, in general such symbols are not even bounded in L1. Hence, from now on we will concentrate on local spaces. As expected, the theory is similar to that for Besov and Zygmund spaces [Ru1], [Ru2], [Bour], [Ma]. First of all, there are better interpolation results, since one uses real interpolation s/2 p on the scale of spaces (I ∆)− M . From Proposition 2.6, − q s1/2 p s2/2 p s (3.18) ((I ∆)− M , (I ∆)− M )θ,r = N , − q − q p,q,r

7 0 L1,δ is another standard notation for OPS1,δ.

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where θ (0, 1) and s =(1 θ)s1 + θs2. Therefore, a linear operator P bounded ∈ si/2 p − s on (I ∆)− Mq for i =1, 2 is also bounded on Np,q,r ( [Trieb2], p. 63). A first partial− result follows easily.

Proposition 3.14. If P OPSm , m R, δ<1,then ∈ 1,δ ∈ s s m (3.19) P : Np,q,r Np,q,r− , s R,r [1, ], 1

To treat non-regular symbols, which do not have good multiplicative properties, we use elementary symbols and follow the approach in [Bour] and [Ma]. Again, the idea is to compare the decomposition in frequency of the symbol with that of the function on which the symbol acts. To do so, we need a couple of preliminary lemmas. For the reader’s convenience we give proofs, although they are very similar to that of Lemma 3.13.

Lemma 3.15. Let fk , k 0, be a sequence of tempered distributions such that for {ˆ } ≥ ˆ n k 1 k+1 some A>0, Supp f0 B(0,A2) and Supp fk ξ R A 2 − < ξ 0.Then ⊂ ⊂{ ∈ | | | }

sk (3.20) fk N s C(A) f0 p + 2 fk p ∞ `r . k k p,q,r ≤ k kMq k{ k kMq }k=1k k X
Proof. Let ϕ0,ψj be the usual (inhomogeneous) Littlewood-Paley partition of { } unity. By hypothesis, ψj is supported on the dyadic shell Dj, while ϕ0 is supported on the ball B(0, 2). Hence, there is an N = N(A) independent of k such that

j+N

ψj(D) fk = ψj(D) fk ,   k ! k=j N X X−  N  ϕ0(D) fk = ϕ0(D) fk . k ! k=0 ! X X Consequently,

N fk N s = ϕ0(D) fk p k k p,q,r k kMq k k=0 ! X X r 1/r j+N

sj p (3.21) + 2 ψj(D) fk Mq   k   k   j 0 k=j N X≥ X−      r 1/r  N j+N  sj C fk p + 2 fk p , ≤ k kMq   k kMq   k=0 j N k=j N X X≥ X−     

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0 since ϕ0(D),ψj(D) is uniformly bounded in OPS1,0. We estimate each part separately:{ }

1/r N N 1/r0 jkr0 ks r fk p 2− (2 fk p ) k kMq ≤  k kMq  k=0 k=0 ! k 0 (3.22) X X X≥  1/r   ks r C (2 fk p ) , ≤  k kMq  k 0 X≥    using H¨older’s inequality with exponent r0 conjugate to r. We then bound the remaining piece:

r 1/r j+N sj 2 fk p   k kMq   j N k=j N X≥ X−    r 1/r  2N  ls (l+j)s = C 2− (2 fl+j p )  k kMq  j 0 " l=0 # (3.23) X≥ X  1/r 2N  ls (l+k)s r C 2− (2 fl+k p ) ≤  k kMq  l=0 k 0 X X≥  1/r   ks r C˜ (2 fk p ) . ≤  k kMq  k 0 X≥    

This is basically Proposition 2.8 in [KY], except that our proof works also for r = (all we used is the triangle and H¨older inequalities). ∞

Lemma 3.16. Let fk , k 0, be a sequence of tempered distributions such that { } ≥ k+1 for some A>0, Supp fˆk B(0,A2 ). Then for s>0, ⊂

sk (3.24) fk N s C(A) 2 fk M p k 0 `r . k k p,q,r ≤ k{ k k q } ≥ k k X Proof. The low-frequency part is taken care of by the previous lemma. Concentrate on the high-frequency part. Again, in view of the hypothesis on Supp ψj,there exists an N = N(A) independent of j such that

(3.25) ψj(D) fk = ψj(D) fk .     k 0 k j N X≥ ≥X−    

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Hence, by the triangle inequality:

sj fk N s 2 fk M p j N `r k k p,q,r ≤k{  k k q } ≥ k k k j N X ≥X−   sj (j+l)s (j+l)s = 2 2− 2 fj+l M p j N `r k{  k k q } ≥ k l N ≥−X (3.26)  ls (j+l)s  2− 2 fj+l M p j N `r ≤ k{ k k q } ≥ k l N ≥−X

ls js 2− 2 fj M p j 0 `r ≤   k{ k k q } ≥ k l N ≥−X  js  C 2 fj M p j 0 `r , ≤ k{ k k q } ≥ k as long as s>0.  We are now ready to prove our main result.

` m Theorem 3.17. Let p(x, ξ) C S1,δ,wherem R, δ [0, 1].Then ∈ ∗ ∈ ∈ (3.27) p(x, D): N s+m N s p,q,r → p,q,r if 0

(3.28) q(x, ξ)= qj(x)ψj (ξ) , j 0 X≥ where qj satisfies j`δ (3.29a) qj C` C 2 , k k ∗ ≤ (3.29b) qj L C, k k ∞ ≤ with C depending on δ and ` but not on j.Bymodifyingqj, if necessary, we can always assume ψj is exactly a Littlewood-Paley function. Set ψk(D)qj = qkj . Then, (3.29a) is equivalent to j`δ k` (3.30) qkj L C 2 2− . k k ∞ ≤

8See Section 3.3.

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Again, the idea is to “match” the decomposition of each qj into qkj with the de- 9 composition of the function f on which q(x, D)actsintoψj(D)f = fj. Therefore, we rewrite the symbol as a sum of three parts, a “low-high”, a “high-high”, and a “high-low” part. Here, high and low refer to the range of frequencies for qj, i.e., k, compared to j:

j+3

q(x, ξ)= qkj (x)+ qkj (x)+ qkj (x) ψj(ξ)   (3.31) j 0 kj+3 X≥ X− X− X = q1(x, ξ)+q2(x, ξ)+q3(x, ξ) . 

We will treat each term separately, starting with the simpler piece q2. j+3 j+3 Since q2(x, D)f = ( qkjfj)in 0,where ( qkj fj) is sup- k 0 k=j 3 S F k=j 3 ported on the ball B(0, 2j≥+4), by Lemma− 3.16, − P P P j+3 js q2(x, D)f N s C 2 qkj fj p `r k k p,q,r ≤ k{ k kMq }k k=j 3 (3.32) X− j+3 js C 2 qkj L fj p `r . ≤ k{ k k ∞ k kMq }k k=j 3 X− But (3.30) implies

j+3 3 j`(δ 1) k` qkj L 2 − 2− C, k k ∞ ≤ ≤ k=j 3 k= 3 X− X− with C depending only on ` and δ.So

js (3.33) q2(x, D)f N s C˜ 2 fj p `r C˜ f N s . k k p,q,r ≤ k{ k kMq }k ≤ k k p,q,r

In the same fashion, since ( qkj fj) is supported on a shell centered at the F k

In the last line, we used the fact that the qj’s are uniformly bounded in L∞. Finally, let us examine q3(x, D)f.Here k>j+3 qkjfj is not supported on balls or shells; hence we cannot directly proceed as above. However, note that in 0, P S

qkj fj = qkj fj , j 0 k>j+3 k>3 j

9 sj Note that in this case: f Ns 2 fj Mp j 0 `r . k k p,q,r ≡k{ k k q } ≥ k

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k and ( j k 4 qkjfj) is now supported on a shell around the frequency ξ =2 . ApplyingF again≤ − Lemma 3.15 yields | | P q3(x, D)f N s = qkjfj N s k k p,q,r k k p,q,r k>3 jseminorm of its symbol in the topology of C S1,δ. Because of the good multiplicative properties of regular∗ symbols, if p(x, ξ) S0 ∈ 1,δ with δ<1, then (3.27) holds for every s R. This result extends Proposition 3.14 to include the case q =1. ∈

Remark 3.19. Since Lemma 3.15 was used to bound q1(x, D)f, the restriction s>0 is not necessary for this type of symbol, as long as `>0. Moreover, the only other ingredient in the proof is (3.29b) or an even weaker estimate, namely

supj k j qkj L∞ < . Consequently, it is also possible to relax the regularity k ≤ k ∞ 0 0 0 assumption to q(x, ξ) L∞ S1,δ (with obvious notation). If q(x, ξ) C S1,δ, P ∈ ∈ ∗ (3.34) just barely fails, while (3.33) still holds for s positive. This observation can r m be formalized further and leads us to study the symbol class S1,δ introduced by Y. Meyer, which has important applications to non-linear partialB differential equations [Tay2]. By using symbol smoothing, we can strengthen our result a bit for the case δ<1, namely we can take s negative (but small) in Theorem 3.17. Essentially, symbol smoothing consists of applying to the symbol standard mollifiers in x. For details, we refer to [Tay1], Vol. 3.

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Corollary 3.20. Under the hypotheses of Theorem 3.17, assume, in particular, that δ<1.Fixγ (δ, 1).Then,for0 <<(γ δ)`, ∈ − +m  (3.38) p(x, ξ): N − N − . p,q,r → p,q,r Proof. Again, we can reduce to the case m = 0. Recall that a symbol p(x, ξ) ` 0 ∈ C S1,δ, δ<1, can be decomposed as follows: ∗ (3.39) p(x, ξ)=p#(x, ξ)+pb(x, ξ) ,

# 0 b ` (γ δ)` with p S1,γ and p C S1−,γ − for a chosen γ (δ, 1). Therefore, ∈ ∈ ∗ ∈ #   (3.40) p (x, D): N − N − , , p,q,r → p,q,r ∀ while b   (3.41) p (x, D): N − N − , p,q,r → p,q,r as long as 0 <<(γ δ)`. In fact, although there is no good calculus for non- b − (γ δ)` ` 0 regular symbols, p (x, D) (I ∆) − belongs nevertheless to C OPS1,γ.  ◦ − ∗ 3.3. Para-differential operators. Para-differential operators are a powerful tool for the study of non-linear differential equations. As before, we start by outlining the main points of para-differential calculus, which was first introduced by J.-M. Bony [Bo] and developed by Y. Meyer [Me]. A rather complete treatment can be found again in [Tay1], Vol. 3. Meyer’s idea is simple, yet ingenious. It consists of writing a non-linear differ- ential operator as a sum of a linear pseudo-differential operator (which will be of type (1,1)) plus a smooth remainder. Suppose F : R R is a smooth (non-linear) function and pick u in some → appropriate space. Let φ0,ψj , j 1, be the usual Littlewood-Paley partition of { } ≥ unity. Set Ψk = φ0 + j k ψj and write F (u)as ≤ (3.42) F (u)=FP(u0)+[F (u1) F (u0)] + [F (u2) F (u1)] + − − ··· with uk =Ψk(D)u. Applying the Fundamental Theorem of Calculus to each dif- ference gives

∞ F (u)= mk(u, x) ψk+1(D)u + F (u0) (3.43) k=0 X = MF (u, x, D)u + F (u0) , where 1 mk(u, x)= F 0(uk + tψk+1(D)u) dt Z0 and F (u0)=R(u) is a smooth remainder (recall ψ0(D) OPS−∞). MF (u, x, D) is called a para-differential operator. ∈

Notation 3.21. Even though, in general, MF does depend on u and F ,itiscus- tomary to write M(x, D) instead of MF (u, x, D). 0 If u L∞,thenMF (u, x, D) OPS , since by the chain rule ∈ ∈ 1,1 α α Dξ M(x, ξ) Cα F 0 L (I) ξ −| | , (3.44) | |≤ k k ∞ h i α β β β α D D M(x, ξ) Cαβ F 00 β 1 u L | | ξ | |−| | , | ξ x |≤ k kC| |− (I)hk k ∞ i h i

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where I is an interval containing the essential range of u. Note that, in particular, F need not be bounded. But F must be smooth in order to have a regular symbol s for M. Therefore, if u N L∞, results from the previous section imply ∈ p,q,r ∩ (3.45) M(x, D): N t N t , t>0 . p,q,r → p,q,r ∀ ` 0 ` Since the operator norm of p(x, D) C OPS1,δ depends only on the C -norm of ∈ ∗ ∗ p( ,ξ) from Theorem 3.17, we obtain the bound · N (3.46) M(x, D) (N t ) KN (F, u)=C F 0 CN u L , k kL p,q,r ≤ k k hk k ∞ i by choosing N>t.Ifs>0, we can apply the above estimates to F (u)itselfand conclude

(3.47) F (u) N s KN (F, u) u N s + R(u) N s ,N>s. k k p,q,r ≤ k k p,q,r k k p,q,r Actually, so far we know R(u) is smooth, but there is no prescribed behavior at ` infinity. Since u is bounded, the chain rule again implies R(u)=F (u0) C for every ` 0, i.e., ∈ ≥ j` ∆jF (u0) L∞ CF 2− , ` 0 . p k ` k ≤ ∀` ≥ s Since L∞ Mq , R(u) Np,q, for all `>0, and Np,q, Np,q,r if `>s. The following⊂ Moser-type∈ ∞ estimate can be obtained∞ by⊂ specializing (3.46) to F (u)=u2, 2 s (3.48) u N s Cs u L u N s , u L∞ N ,s>0 . k k p,q,r ≤ k k ∞ k k p,q,r ∀ ∈ ∩ p,q,r

Indeed, in this case F 0 L∞ + F 00 L∞ u L∞ 2 u L∞ , and no other term arises because higher-orderk k derivativesk k ofhkF kare zero.i≈ hk k i Let us now consider a function of several variables F (u)=F (u1,...,um). Then, expanding F as before gives m (3.49) F (u)= MF,i(u, x, D)u + F (ψ0(D)u) , i=0 X where

∞ i (3.50) MF,i(x, ξ)= mk(x) ψk+1(ξ) k=0 X and 1 i mk(x)= ∂iF (Ψk(D)u + tψk+1(D)u) dt . 0 i Z Hence, as long as each u belongs to L∞, all the results of the one-dimensional case extend. In particular, for F (u, v)=u v, (3.49) becomes · (3.51) u v = M˜ F (u, x, D)v + M˜ F (v, x, D)u + ψ0(D)uψ0(D)v, · with 1 (3.52) M˜ (u, x, ξ)= [Ψ (D)u + ψ (D)u] ψ (ξ) . F k 2 k+1 k+1 k 0 X≥ The equivalent of (3.44) is k` M˜ F (u, ,ξ) ` C 2 u L ψk+1(ξ) k · kC ≤ k k ∞ (3.53) k 0 X≥ ` C u L ξ , ≈ k k ∞ h i

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` k` since ξ 2 on Supp ψk+1. Therefore, a Moser-type estimate follows: h i ∼ (3.54) u v N s C( u L v N s + u N s v L ) . k · k p,q,r ≤ k k ∞ k k p,q,r k k p,q,r k k ∞ It is interesting, especially in view of applications to the Navier-Stokes equa- 10 s tion, to establish when the space Np,q,r is an algebra. Given the discussion above, s it is enough to find when N , L∞. p,q,r → Recall first the situation for Besov spaces [Ru1]. For 0 0, 0 n/p, 1

s Corollary 3.22. The space Np,q,r is a Banach algebra for (1) s>n/p, 1 q p< ,ifr (1, ],or (2) s n/p, 1 ≤ q ≤ p<∞,ifr =1∈ . ∞ ≥ ≤ ≤ ∞ Another way of decomposing a product of two functions is the paraproduct of J.-M. Bony [Bo]. We follow [Tay2] and define it as

(3.59) π(v, w)= (Ψk 1(D)v)(ψk+1w) . − k 1 X≥ So, π(u, v) is obtained by applying symbol smoothing directly to the multiplication ` operator by v, mv.Whenv C , `>0, one has ∈ (3.60) v w = π(v, w)+ρv(x, D)w, · ` ` with ρv(x, ξ) C S− . Note that, by the results of the previous section, ∈ 1,1 s ` s ρv(x, D):N − N p,q,r → p,q,r only if 0 0 [Tay2], which allows us to improve slightly upon (3.60). ∈

10Recall the non-linearity in the Navier-Stokes equation is quadratic.

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` s Corollary 3.23. Let F beasmoothfunction.Ifu C Np,q,r,withs>0, `>0 , the following decomposition holds: ∈ ∩

(3.62) F (u)=π(F 0(u),u)+R˜(u) , where R˜(u) N s+` . ∈ p,q,r In particular, the best approximation to u2 is not π(u, u), but rather 2 π(u, u).

Proof. Write

F (u)=π(F 0(u),u)+(MF (x, D)u π(F 0(u),u)) + R(u) , − ˜ with R(u)=(MF (x, D)u π(F 0(u),u))+R(u). But R(u)issmoothand(MF (x, D)u − s+` π(F 0(u),u)) = QF (x, D)u belongs to N because of (3.61). − p,q,r 

The remainder term ρu in (3.60) can be decomposed further to obtain a form of paraproduct that matches closely the expansion (3.31) for an elementary symbol and emphasizes the symmetry between u and v.Set

(3.63) u v = Tu v + Tv u + R(u, v) , · where

(3.64a) Tu v = Ψj 2(D)uψj(D)v, − j 1 X≥ (3.64b) R(u, v)=Ru v = ψk(D)uψj(D)v. k j 1 | −X|≤ Note that both Tu and Ru have symbols of the form discussed in Remark 3.19. Precisely, Tu is a symbol of “type” q1, while Ru is of “type” q2. Here, qkj = ψk(D)u,ifk j 2orj 1 k j + 1 respectively, and ≤ − − ≤ ≤ zero otherwise (in the notation of Section 3.2). Therefore, whenever the qkj ’s are 0 uniformly bounded in L∞,thatisu C , ∈ ∗ s s s s  (3.65) Ru : N N ,Tu : N N − , p,q,r → p,q,r p,q,r → p,q,r µ for all >0, as long as s>0. In fact, a similar analysis shows that, if u C− , µ ∈ ∗ µ>0, Tu and Ru belong to the operator class OPS1,1,sothat

(3.66) T v s µ C u µ v s µ>0, u Np,q,r− C− Np,q,r k k ≤ k k ∗ k k ∀ (3.67) R v s µ C u µ v s µ ,s>µ. u Np,q,r− C− Np,q,r R k k ≤ k k ∗ k k ∀ ∈ We can then extend the Moser estimates (3.54):

(3.68a) u v s µ C( u µ v s + u s v µ ), µ>0,s>µ, Np,q,r− C− Np,q,r Np,q,r C− k · k ≤ k k ∗ k k k k k k ∗ ∀ (3.68b) u v s+` n/p C( u N s v N ` ), if s + `>n/p. k · kNp,q,r− ≤ k k p,q,r k k p,q,r For s = n/p,inparticular,

(3.69) u v n/p  C u n/p v n/p , >0 , k · kNp,q,r− ≤ k kNp,q,r k kNp,q,r ∀ n/p which shows how close Np,q,r is to a Banach algebra.

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4. Extension to Manifolds The aim of this section is to define the BM spaces on compact manifolds. In hydrodynamics, boundary value problems are a rather natural setting. At the same time, due to the incompressibility constraint, non-local effects are likely to become relevant. Even when boundaries are not present, curvature terms may not be negligible. We will concentrate on the simplest case, that of a (smooth) compact manifold M without boundary. We plan to address the question of manifolds with boundary in future work. It will be necessary to use pseudo-differential calculus both on BM and Morrey s spaces. Consequently, we are able to consider only the inhomogeneous spaces Np,q,r, q>1. 4.1. Main definition. A natural way to extend the definition of a function space X from Rn to compact manifolds is by partitions of unity and coordinate charts. Then any property that is localizable also extends. There are clearly two steps to this process: n n n (1) given f X(R )andφ C0∞(R ), show φ f X(R ); ∈ ∈ · ∈ (2) given a diffeomorphism χ of Rn, show that the topology of X(Rn)isinvari- ant under the action of χ. Here, as throughout the section, we implicitly identify a coordinate chart U on M with its image in Rn and, consequently, χ represents a certain coordinate trans- formation. By refining the charts on M, if necessary, one can reduce to considering a diffeomorphism that is linear outside a compact set K. In particular, the Jacobian of the transformation ∂χ/∂x is bounded by some constant A>0. | | Notation 4.1. Since we are primarily concerned with the inhomogeneous space s Np,q,r,here ψj j 0 is a Littlewood-Paley partition of unity as in (3.15), i.e., ψj is { } ≥ n j 2 j supported on the dyadic shell Dj = ξ R 2 − ξ 2 for j>0, but ψ0 is supported on the ball B(0, 2). { ∈ | ≤| |≤ } p Recall first how the Morrey spaces Mq ,1 q p< , are defined on manifolds. p ≤ ≤ ∞ It is enough to prove that Mq is invariant under the action of χ, since it is immediate p that the multiplication operator mφ,wheremφf = φf, is bounded on Mq .Aneasy change of variable gives 1/q

p 1/q n/p n/q q (4.1) χ∗ f Mq A sup R − f(y) dy , k ◦ k ≤ 0

p 1/q n/p n/q q p χ∗ f Mq NA sup R − f(y) dy C f Mq . k ◦ k ≤ 0

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s Next, we proceed similarly in identifying Np,q,r(M). However, the situation is not as straightforward, because the Littlewood-Paley theory does not carry over directly to manifolds. Indeed, each ψj(D) is inherently a non-local operator. Moreover, only the principal symbol of a pseudo-differential operator is a well-defined covariant object. We are forced to deal with asymptotic expansions and equivalence classes of symbols.

p p Remark 4.3. Note that, obviously, Mq (M) q(M), if M is compact. On the ≈Ms n s n other hand, compactly-supported elements of Np,q,r(R )and p,q,r(R )donot N necessarily coincide. Take, for example, f = χB(0,1), the characteristic function of the unit ball in Rn. We can estimate its Morrey norm as follows. First,

jn j ψj(D)f(x)= 2 ψˇ(2 y) dy = ψˇ(z) dz , j ZB(x,1) ZB(x,2 )

where ψ = ψ1.Sinceψˇ is rapidly decreasing at infinity, the largest contribution to ψj(D)f p comes from picking x =0,sothat k kMq

p ˇ (4.3) ψj(D)f Mq C ψ(z) dz . k k ≈ | j | ZB(0,2 ) Choose s negative. Then

1/r js r (4.4) f N s = (2 ψj(D)f p ) + ψ0(D)f p k k p,q,r  k kMq  k kMq j>0 X   j s (4.5) C ψˇ0 1 2− | | + ψ0(D)f p < . ≤ k kL   k kMq ∞ j>0 X   On the other hand, if s >nthe low-frequency part (j<0) determines a divergent contribution to the norm| | of s . As a matter of fact, since ψˇ(0) = ψ(ξ) dξ > Np,q,r Rn 0, ψˇ is strictly positive on some neighborhood of the origin. (4.3) then gives R j n (4.6) ψj(D)f p C 2−| | , k kMq ≥ for j negative large enough (say j

1/r 1/r r js r j ( s n) (4.7) f s (2 ψj(D)f M p ) C 2| | | |− . k kNp,q,r ≥  k k q  ≥   j

Let U be a coordinate patch on M and φ C∞(M) be supported on U.The ∈ 0 multiplication operator mφ is a pseudo-differential operator in OPS1,0. Therefore, s it is bounded on Np,q,r by virtue of Proposition 3.14, and we can restrict to functions f supported on coordinate patches.

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The next necessary ingredient is to prove that the action of a diffeomorphism s n preserves the topology of Np,q,r(R ). We have to estimate

js 2 ψj(D)(χ∗f) p n `r . k{ k kMq (R )}k p n But, since χ∗ maps Mq (R ) to itself, it is equivalent to showing instead that

js 1 (4.8) 2 χ∗− ψj(D)χ∗f p n `r C(χ, s, p, q, r) f N s . k{ k kMq (R )}k ≤ k k p,q,r The advantage is that the action by conjugation of a diffeomorphism on a pseudo- differential operator (at least of type (1,δ)withδ<1/2) can be treated using Egorov’s theorem, and there is a complete asymptotic expansion of the resulting symbol (see [Tay4] or [Tr] for a proof). 1 For convenience, we denote the symbol of χ∗− ψj(D)χ∗ with pj(x, ξ). The fol- lowing expansion holds:

1 α α (k) (4.9) pj(x, ξ) ∂ξ Dz qj(x, ξ, z) rj (x, ξ) , ∼ α! z=x ∼ α k X X where ∂χ 1 (4.10) q (x, ξ, z)=ψ (H(x, z)t 1ξ) φ(x, z) , j j − ∂z H(x, z)

| | with H(x, z) defined by

(4.11) χ(x) χ(z)=H(x, z)(x z) , − − (k) and rj (x, ξ) is obtained by collecting all the terms in the expansion of the same order, namely α = k. Since clearly H(x, x)=∂χ/∂x, which is non-zero by | | hypothesis, H(x, z) is well-defined and smooth near the diagonal in Rn Rn.The function φ is supported in a neighborhood of x = z and acts as a regularizing× factor, as H may not be defined everywhere. The error introduced in this way is a symbol supported in (x, z) away from the diagonal; thus it is of order and does not contribute to the asymptotic expansion. −∞ Note that, even though each term in (4.9) is the symbol of a smoothing operator, its norm depends on j, while we want uniform estimates. Since ψj is supported on j (k) k the shell Dj where ξ 2 , it follows that rj (x, ξ) j 0 is bounded in S1−,0 and h i∼ { } ≥ jl (k) l k 0 2 rj (x, ξ) j 0 is O(2 − )inS1,0 for 0 l k. { } ≥ ≤ ≤ Let us concentrate first on the principal symbol of pj(x, ξ). Write ∂χ t 1 (4.12) p (x, ξ)=ψ (−x) ξ + p(1)(x, ξ)=r(0)(x, ξ)+p(1)(x, ξ). j j ∂x j j j   ∂χ t 1 Again, the ξ-support of ψ (−x) ξ can be covered by a finite number N, inde- j ∂x   ∂χ pendent of x, of shells D , as the Jacobian matrix is constant outside a compact m ∂x set. Therefore,

j+N ∂χ t 1 ∂χ t 1 (4.13) ψ (−x )ξ = ψ (−x) ξ ψ (ξ) . j ∂x j ∂x m   m=j N   X−

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t 1 ∂χ − 0 But ψj (x) ξ j 0 is bounded in S1,0. Accordingly, these symbols define { ∂x } ≥   s operators Qj that are uniformly bounded in Np,q,r and

j+N j+N

(4.14) Qjf p Qj ψk(D)f p C ψk(D)f p , k kMq ≤ k ◦ kMq ≤ k kMq k=j N k=j N X− X− with C independent of j.Furthermore, (4.15) j+N js (j k)s ks 2 Qjf p `r C 2 − 2 ψk(D)f p `r C˜ f N s , k{ k kMq }k ≤ k{ k kMq }k ≤ k k p,q,r k=j N X− by the Young-Hausdorff inequality (cf. proof of Theorem 3.17). (1) Next, we look at the residual term pj .From(4.9),

n (1) (2) pj (x, ξ)= ∂ξi Dzi qj(x, ξ, z) + pj (x, ξ) z=x (4.16) i=1 X (1) (2) = rj (x, ξ)+pj (x, ξ) ,

2j (2) 0 (1) with 2 pj (x, ξ) bounded uniformly in j in S1,0. rj can be written as a finite sum of terms of the form

(4.17) hj(x) gj (x, ξ) ,

with hj(x) uniformly bounded in L∞ and gj smooth with compact support in 0 both x and ξ, uniformly bounded in S1,0.Moreover,sincegj is basically either j t 1 j j t 1 2− ψj00(H(x, z) − ξ)(2 ξi)(i =1, ,n)or2− ψj0 (H(x, z) − ξ), estimates similar ··· (0) to (4.13) apply. Hence, an analysis parallel to that of rj can be carried out yielding

js (1) (4.18) 2 r ( ,D)f p `r C f N s . k{ k j · kMq }k ≤ k k p,q,r

Remark 4.4. Note that, when differentiating qj(x, ξ, z) with respect to z, factors of ξi are produced. However, they do not change estimates of the symbol for large j, j since they always appear in the combination 2− ξi,whichisO(1). Iterating this process takes care of any finite number of terms in the expansion (4.9). At the k-th step,

(l) (k) (4.19) pj(x, ξ)= rj (x, ξ)+pj (x, ξ) , l k 1 ≤X− (l) where all the rj (x, ξ) share a form similar to (4.17) and satisfy an estimate similar jl (k) l k to (4.18), while the remainder is such that 2 pj (x, ξ) j 0 is bounded in S1−,0 { } ≥ if 0 l k.So,ifk is large enough (k>max(s +1,n/p+2)for s 0and k>n/p≤ +2≤ s if s<0) and l max(s, 0)+1,wehavethefollowingestimate:≥ − ≤ jl (k) jl (k) jl (k) p n/p+1 2 pj (x, D)f Mq 2 pj (x, D)f L 2 pj (x, D)f (4.20) k k ≤ k k ∞ ≤ k kNp,q,r Cl,p,q,r f n/p+1 (k l) Cl,p,q,r f N s , ≤ k kNp,q,r − − ≤ k k p,q,r

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p by Sobolev embedding. (Recall L∞ Mq .) Then, choose l =max(s, 0) + ,0< <1toobtain ⊂ js (k) j(l ) (k) 2 p (x, D)f p `r 2 − p (x, D)f p `r k{ k j kMq }k ≤k{ k j kMq }k j jl (k) (4.21) 2− ` 2 p (x, D)f p `r ≤k{ }k ∞ k{ k j kMq }k Cs,p,q,r f N s . ≤ k k p,q,r (4.8) now follows from (4.15), (4.18), and (4.21).

s Definition 4.5. A distribution f on M belongs to Np,q,r(M) if, for every coordinate s n patch U on M and φ C0∞(U), φf Np,q,r(R ). Then, we set for each f s ∈ ∈ ∈ Np,q,r(M):

(4.22) f s =sup φ f s n , Np,q,r(M) α Np,q,r(R ) k k φα { α k k } { } X where the supremum is taken over all possible representations of f in terms of partitions of unity φα supported on coordinate patches Uα. { } 4.2. Equivalent norms. When considering spaces on manifolds, it is useful to work in a covariant setting. Clearly, the operator D =(1/i)∂ cannot be used. A covariantly defined operator that is intimately related to the geometry of a manifold is the Laplace operator ∆. It is, therefore, natural to try to define the topology of s Np,q,r(M) in terms of functions of √ ∆, an operator of order one such as D. As before, we assume that M is smooth,− compact, without boundary. We assume further that M is Riemannian with metric g =[gij ]. We shall write ∆g for the Laplace operator associated to the metric g, i.e., 1/2 jk 1/2 (4.23) ∆g =(detg)− ∂jg (det g) ∂k,

while ∆ still denotes the Euclidean Laplacian, and we will identify ∆g with its local representation on Rn via a coordinate system. Families of operators of the form ρ(δ ∆g), where ρ is an even, rapidly de- creasing function, and δ (0, 1], are called−approximate identities if ρ(ξ)=1ina neighborhood of the origin.∈ They have severalp applications to the theory of partial differential equations on manifolds (see [ST], for example).

Remark 4.6. Let ψj be the Littlewood-Paley partition of unity used in Section 4.1. { } j Recall that, by construction, ψj = ψ(2− ξ)forj>0, where ψ ψ1.Inparticular, ≡ there is no loss of generality in assuming that ψ is radial, i.e., ψ(ξ)=ψ˜( ξ )for | | some function ψ˜ on R. With abuse of notation, we identify ψ with ψ˜. Then, ψ j corresponds to ρ above and 2− to δ. The main result of this section is the following theorem.

Theorem 4.7. Let s R, 1

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We follow Schrader and Taylor [ST] in part, but our derivation is admittedly very similar to that of Seeger and Sogge [SeSo] for Besov and Triebel-Lizorkin spaces. However, their argument is simplified owing to the nice properties of Lp spaces. In particular, they can reduce to considering a C∞ function and do not have to bound remainder terms. On the other hand, smooth functions are not dense in BM spaces. The core of the proof again consists of deriving careful pseudo-differential esti- mates. Unfortunately, operators on manifolds correspond to equivalence classes of symbols, as already mentioned. Hence, we must deal with infinite expansions at each step. We use the following asymptotic expansion derived in [ST]:

2k (2k) (4.25) σ(ρ(δ ∆g)) δ ρ (δ ξ x) pk(x, δξ) . − ∼ | | k 0 p X≥ (2k) Here, pk(x, δξ) is a classical symbol of order k, ρ is the 2k-th derivative of ρ with ij 1/2 respect to its argument, and ξ x =( g (x)ξiξj) is the norm of the covector | | i,j (x, ξ) in the cotangent bundle T (M). ∗ P The expansion above is obtained by formally writing ρ(δ√ ∆) in terms of its cosine transform (ρ is an even function) −

1 ∞ (4.26) ρ(δ ∆g)u(x)= ρˆδ(t)cos(t ∆g)u(x) dt , − 2π1/2 − Z−∞ p p where ρδ(τ)=ρ(δτ). Recall that cos(t ∆g) is (half of) the propagator for the wave equation and, consequently, its kernel− satisfies the finite propagation speed p property. It follows that the Schwartz kernel of ρ(δ ∆g) is rapidly decreasing, as δ 0, outside an arbitrarily small region around− the diagonal in M M. → p × Therefore, one can limit integration in (4.26) to t T0, T0 small, by localizingρ ˆδ to a neighborhood of the origin, the error being a| smoothing|≤ operator that is rapidly decreasing in δ and, thus, negligible. Moreover, one can work in local coordinates. In this situation, it is convenient to use geometrical optics methods and station- ary phase approximation, which leads to (4.25). In particular, the appearance of only even-order derivatives is due to an extra symmetry that the phase functions and amplitudes possess. To proceed with the proof of the theorem, we first restrict to a coordinate patch U, which amounts to proving js js (4.27) 2 ψj( ∆g)(φf) p `r C 2 ψj( ∆g)f p `r k{ k − kMq (M)}k ≤ k{ k − kMq (M)}k 0 for all φ C∞(U).p Since the multiplication operator mφ pOPS , clearly, ∈ 0 ∈ 1,0 js js (4.28) 2 φ(ψj( ∆g)f) p `r C 2 ψj( ∆g)f p `r . k{ k − kMq (M)}k ≤ k{ k − kMq (M)}k Then we switch thep order of the product and attempt top bound the commutator [ψj( ∆g),mφ] appropriately. Alternatively, notice that mφ and ψj( ∆g)are both selfadjoint− with respect to g (φ canalwaysbechosentobereal).Therefore,− p p ψj( ∆g) mφ =(mφ ψj( ∆g))∗. If p−(x, D)◦ is a pseudo-differential◦ − operator with symbol p(x, ξ), there is an explicit p p integral expression for the action of p(x, D)∗ in local coordinates:

1 i(x y) ξ (4.29) p(x, D)∗f(x)= n p(y,ξ)∗ f(y) e − · dydξ , (2π) n n ZR ZR

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where the integration must be performed in the given order (or else consider the Fourier Transform acting on distributions). Here, p(y,ξ)∗ is not a symbol in the usual sense. These more general objects are sometimes called amplitudes and are very useful in treating adjoints and products of pseudo-differential operators (see [Tr] and [Tay4], for example). Therefore, we can rewrite ψj( ∆g) mφf(x)as − ◦ (mφ ψj ( ∆g))∗f(x) p ◦ − p1 i(x y) ξ = n σ(mφ ψj( ∆g))∗(y,ξ) f(y) e − · dydξ (4.30) (2π) n n ◦ − ZR ZR 1 p i(x y) ξ = n σ(mφ ψj( ∆g))(y,ξ) f(y) e − · dydξ , (2π) n n ◦ − ZR ZR because the symbols are all real. p Remark 4.8. In general, the symbol of the product q(x, D) of two operators pj(X, D), j =1, 2, is given by an infinite expression of the form α i| | α α (4.31) q(x, ξ) D p1(x, ξ) D p2(x, ξ) . ∼ α! ξ x α 0 |X|≥ By contrast, q(x, ξ) becomes simply the product of the symbols, whenever p1(x, ξ) is actually independent of ξ or p2(x, ξ) is independent of x.

In particular, since σ(mφ)(x, ξ)=φ(x) does not depend on ξ,

(4.32) σ(mφ ψj ( ∆g))(y,ξ)=φ(y) σ(ψj ( ∆g))(y,ξ) . ◦ − − So finally, after identifyingpf and φ with their expressionp in local coordinates,

ψj( ∆g) mφ f p k − ◦ kMq 1 i( y) ξ p = φ(y)σ(ψ ( ∆ ))(y,ξ) f(y) e dydξ p n j g ·− · Mq (2π) k n n − k ZR ZR 1 p φ L ψj( ∆g)f p , ≤ (2π)n k k ∞ k − kMq which proves (4.27). p We split the proof of Theorem (4.7) into two distinct parts. Let A be the 11 n operator obtained by transporting ∆g to R . − n Lemma 4.9. Let s R, 1

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j (1, 0) and 2− appears in combination with ξ. Therefore, the term of order k in k j(l 2k) (k l) (4.34) is uniformly bounded in S1−,0 , with respect to j,andO(2 − )inS1−,0 − for 0 l k. To prove the lemma, it is then enough to show that each term in the above≤ expansion≤ is a compactly-supported function and that the ξ-support is still essentially a dyadic shell. We can assume that gij is the identity outside a compact set in Rn. The usual (2k) j argument shows that Supp ψ (2− ξ x) is covered by a fixed number of shells { | | } Dm, so that again, as in (4.13), one writes

l+N (2k) j j (2k) j j (4.35) ψ (2− ξ x)pk(x, 2− ξ)= ψ (2− ξ x)pk(x, 2− ξ) ψl(ξ) , | | | | j=l N X− with N independent of x and j. j (2k) j j Let Qk be the operator with symbol ψ (2− ξ x)pk(x, 2− ξ). Then, an analysis parallel to (4.14) and (4.15) yields | |

js j (4.36) 2 Q f p `r C(s, p, q, r) f N s . k{ k k kMq }k ≤ k k p,q,r Again, we obtain a bound on any finite number of terms in the expansion (4.34). To conclude the proof, it is therefore sufficient to show that the norm of the remainder is weak enough, i.e., we set

j 2jk (2k) j j j (4.37) σ(ψ(2− A)) = 2− ψ (2− ξ x) pk(x, 2− ξ)+rL(x, ξ, 2− ) , | | 0 k L 1 ≤X≤ − and we prove that

js j (4.38) 2 rL(x, D, 2− )f p `r C(s, p, q, r, L) f N s , k{ k kMq }k ≤ k k p,q,r js j j(s+l 2L) l L for L large enough. But, since 2 rL(x, ξ, 2− )isO(2 − )inS − for 0 l 1,0 ≤ ≤ L, (4.38) follows from estimates similar to (4.20) and (4.21).  The opposite inequality is more complex. Informally, it is true because the j j principal symbol of ψ(2− ∆g), ψ(2− ξ x), enjoys properties similar to those of − | | ψ(2 jξ). However, a careful analysis of the terms of lower order is necessary. − p Lemma 4.10. Under the hypotheses of Lemma 4.9, it also follows that

js js (4.39) 2 ψj(D)f p n `r C 2 ψj(A)f p n `r . k{ k kMq (R )}k ≤ k{ k kMq (R )}k Proof. The basic idea is somehow to “reverse” (4.35). For each fixed x, ψj( ξ x) is a partition of unity. Therefore, { | | }

l=j+N(x)

(4.40) ψj(ξ)= ψj (ξ) ψl( ξ x) . | | l=j N(x) X− In principle, N(x) varies with x Rn. However, since the metric gij is the identity outside a compact set, an argument∈ similar to that leading to (4.13) shows N(x) N for some fixed number N. We would then like to replace symbols by operators and≤ use uniform estimates on the norm of ψj(D). Using the notation introduced in the l previous lemma, let us call Q0 the operator whose symbol is ψl( ξ x). Unfortunately, l | | ψj(ξ)ψl( ξ x) is not the symbol of the product ψj(D) Q , but one can get around | | ◦ 0

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this problem again by using amplitudes instead of symbols. Since ψj(D)isself- adjoint (with respect to the Euclidean metric on Rn), one deduces similarly to (4.30): l l ψj(D) Q f(x)=(Q ∗ ψj(D))∗f(x) ◦ 0 0 ◦ 1 l i(x y) ξ = n σ(Q0∗ ψj(D))(y,ξ) e − · f(y) dy dξ (4.41) (2π) n n ZR ZR 1 l i(x y) ξ = n ψj(ξ) σ(Q0∗)(y,ξ) e − · f(y) dy dξ, (2π) n n ZR ZR where the last line follows from (4.31), this time because the symbol of ψj(D)is independent of x.Moreover, l σ(Q ∗)(y,ξ) ψl( ξ y) , 0 ∼ | | 12 1 l 0 up to a residual rl (x, ξ), which is O(2− )inS1,0. Using (4.40) and (4.41) along with the above result, we finally get

1 i(x y) ξ ψj(D)f(x)= n ψj(ξ) e − · f(y) dydξ (2π) n n ZR ZR l=j+N 1 i(x y) ξ = n ψj(ξ) ψl( ξ x) e − · f(y) dydξ (2π) n n | | l=j N R R X− Z Z l=j+N 1 l 1 i(x y) ξ = n ψj(ξ)[σ(Q0∗)(y,ξ) rl (y,ξ)] e − · f(y) dydξ (2π) n n − l=j N R R X− Z Z l 1 = ψj(D) Q f(x) ψj(D) r˜ (x, D)f(x) . ◦ 0 − ◦ l l By (4.34) replacing Q0 with ψl(A) introduces additional lower-order terms. In any case, we have l=j+N 1 j (4.42) ψj(D)f p ψl(A)f p + R ( ,D,2− )f p , k kMq ≤ k kMq k · kMq l=j N X− 0 1 as ψj(D) is bounded uniformly in j in OPS1,0,whereR is obtained by collecting 1 j all the terms of lower order. Note that R (x, ξ, 2− ) still has compact support essen- j 1 j tially “centered” around the support of ψl( ξ x) and, as before, 2 R (x, ξ, 2− ) j 0 0 | | { } ≥ is bounded in S1,0. Hence, the same analysis performed in (4.40) through (4.42) 1 j can be carried out here. There is a small complication, namely R (x, ξ, 2− )ψl( ξ x) 1 j l | | is not the symbol of the product of the operators R (x, D, 2− )andQ0 no matter 2 what the order of the product is; so the equivalent of (4.41) is true mod S1−,0 .But the error contributes only to the calculation at the next order. After L steps, we obtain the following estimate:

l=j+ML L j (4.43) ψj(D)f p ψl(A)f p + R ( ,D,2− )f p , k kMq ≤ k kMq k · kMq l=j ML X− L j jL 0 where ML depends on L, but not on j,andR (x, ξ, 2− )isO(2− )inS1,0.As in the previous lemma, it is enough to establish that the norm of the residual is sufficiently weak to conclude the proof.

12 α l It is precisely α >0(i| |/α!) DξDx ψ(2− ξ x). | | | | P

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L j p First, we show that R ( ,D,2− )f Mq is weaker than a Sobolev norm with an appropriate large negativek index.· Sobolevk norms are a natural choice, because ξ j h i has roughly size 2 on Supp RL. L j L j Now, R (x, D, 2− )f ,sinceR (x, D, 2− ) is smoothing and f 0;so H¨older’s inequality gives ∈S ∈E (4.44) L j n/p n/q 1/t L j p m R ( ,D,2− )f Mq sup R − Vol(B(x0,R)) R ( ,D,2− )f L , k · k ≤ 0

1 m m 2 ∞ j(α L) 2 (4.49) f α L n 4 − ψ (D)f dx . Hm− ( ) j k k R ≈  n  | |   R j=0 Z X     Then pick L>2 s +1+n(1/q 1/m):  | | − ∞ j(α L) f α L C 2 − ψj(D)f Lm k kHm− ≤ k k (4.50) j=0 X j(α L s)+jn(1/q 1/m) js jn(1/q 1/m) C 2 − − − r 2 − − ψ (D)f m r , ` r 1 j L ` ≤ k{ }k − k{ k k }k by the triangle and H¨older’s inequalities. Finally, since ψj(D)f is rapidly decreasing and m p q, by Berstein’s inequality ≥ ≥ jn(1/q 1/m) 2− − ψj(D)f Lm( n) C ψj(D)f Lq( n) C ψj(D)f q n (4.51) k k R ≤ k k R ≤ k kMq (R ) C ψj(D)f p n . ≤ k kMq (R ) The lemma now follows from (4.43). 

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q q To show that the L -norm is comparable to the Morrey norm in Mq for a rapidly decreasing function g, pick a ball of large radius around the origin, say B(0,K), where the radius is so big that g Lq(B(0,K)c) is smaller than any given . Then, cover B(0,K) with a finite numberk k of balls of given radius R (0, 1). Recall that, p q ∈ in general, g M is only locally in L . Note also that, in the case of g = ψj(D)f, ∈ q because ψˇj are concentrated on balls of shrinking size, the radius K can be taken to be independent of j, yielding uniform estimates.

Remark 4.11. As observed in [SeSo], one can replace ∆g in (4.24) with any first-order elliptic (classical) pseudo-differential operator−P that is positive and selfadjoint on L2(M). p We conclude this section with one final intrinsic characterization of BM spaces. s n Recall Np,q,r(R ) can be characterized as a real interpolation space (Proposition 2.6):

s n s1/2 p n s2/2 p n (4.52) Np,q,r(R )= (I ∆)− Mq (R ), (I ∆)− Mq (R ) . − − r,θ  s  It is reasonable to expect a similar property will hold for Np,q,r(M), given its def- inition in terms of local charts. Observe that we can form the interpolation space s1/2 p s2/2 p (I ∆g)− M (M), (I ∆g)− M (M) . It is only a matter of recogniz- − q − q r,θ ing this space appropriately.
Proposition 4.12. For s R, r [1, ] and 1

5. Applications to Semi-linear Parabolic Equations In [KY], Kozono and Yamazaki studied solutions to a class of semi-linear para- bolic equations with initial data in certain BM spaces on Rn. This class includes the Navier-Stokes equation (NS for short).

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Independently, Cannone [Can] developed the notion of adapted spaces for NS, combining Littlewood-Paley theory with Bony’s paraproduct. This concept gives a rather general criterion under which existence and uniqueness can be established by simple contraction mapping or Picard iteration. Then, Karch [Kr] pushed Can- none’s approach further, in a more abstract setting, by introducing the spaces BEα. In this section, we compare and generalize these apparently different results, using the material previously developed. In particular, all the conclusions obtained in [KY] can be recovered from the more general theory of adapted and BEα spaces. We also study the same class of equations on compact manifolds. As pointed out by Ebin and Marsden [EM], the Navier-Stokes equation must be modified to take curvature into account. Sections 5.1 – 5.3 recall known results and Sections 5.4 – 5.5 present new results.

5.1. Background. Semi-linear parabolic equations on Rn can be fundamentally treated using methods for ordinary differential equations. In particular, an Initial Value Problem (IVP for short) can be recast in the form of an integral equation with coefficients in a Banach space. This formulation suggests using fixed-point theorems. Let A be a positive semi-definite operator on a Banach space X, such that A generates a C0 semigroup U(t), t 0, on X. Consider the following IVP: − ≥ (5.1) ut + Au = F (u),u(0) = a, where F :[0,T] X is continuous, is an open subset of X,anda X.We use the notation×O→u(t)(x)=u(t, x), x XO. ∈ ∈ Example 5.1. Take A = ∆, F (u)=P div(u u), X = L2, = Hs, s>5/2, n =3.Then(5.1) is the classical− formulation of⊗ NS (see SectionO 5.3). In this abstract setting, boundary conditions or decay at infinity for unbounded domains are implicit in the choice of the function space X. Definition 5.2. Afunctionu C([0,T],X) that satisfies the integral equation ∈ t (5.2) u(t)=U(t)a + U(t s)F (u(s)) ds, 0 t T, − ≤ ≤ Z0 is called a mild solution to (5.1). In general, the integral formulation is weaker than the initial IVP, because we only need U(t )F (u( ) L1([0,T],X). This is especially true if U(t)hassmooth- ing properties−· as in the· heat∈ semigroup. However, if F is Lipschitz, X is reflexive, and a (A), then any mild solution to (5.2) is actually a strong solution to (5.1). Moreover,∈D it is a theorem of J. M. Ball that, in this case, every weak solution is unique and it is, in fact, a mild solution (see [Pa]). Unfortunately, this result does not help in establishing uniqueness of Leray-Hopf solutions for NS, since we do not know if F is globally Lipschitz. But it motivates looking for mild solutions for “rough” initial data, even when the data do not belong to L2 by means of splitting techniques (see, for example, [Lio]). Note that the heat semigroup is not strongly continuous in these spaces. Hence, not all mild solutions are also strong solutions, and the initial value is attained only in a weak sense, which will be made precise. Typically, the data are distributions taken from a scale of spaces and there exists a range of indices bounded from below for which the Banach contraction theorem

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applies to the integral equation, yielding local-in-time, unique solutions. The lower bound corresponds to the so-called limit spaces, characterized by invariance of the norm under the natural scaling of the corresponding equation. In this case, it is necessary to impose an extra “smallness” condition on the initial data and (global) existence is then obtained by direct analysis of Picard iterates. But uniqueness holds only in an appropriate subspace of the solution space. There is an extensive literature13 on the Navier-Stokes equation with initial data in Lp (e.g. [FK], [Ka1]) and Morrey Spaces (e.g. [Fe], [Ka2], [Ka3], [GM], [GMO], [Tay3]). Hydrodynamics in Morrey spaces is interesting for several reasons. For example, the theory extends to include Radon measures µ ˜ p, which can model ∈ M vorticity supported on singular sets of Rn,e.g. vortex rings and vortex sheets.Also, in the specific context of adapted spaces, we have the following result (in dimension n =3). Proposition 5.3 ([LMR]). Suppose E is an adapted space to NS such that (1) E, L2 , → loc (2) for all λ>0, the dilation f fλ is continuous on E. 3 t∆ → Then E, M2 and limt 0 √t e f L =0. → → k k ∞ Recall that an adapted space E is characterized by the property

(5.3) ψj(D)(fg) E ηj f E g E , j Z, k k ≤ k k k k ∀ ∈ where j (5.4) 2−| | ηj < . ∞ j X∈Z The above conditions ensure that the right-hand side of (5.2) defines a Lipschitz map locally-in-time and, hence, that a short-time, unique solution to NS exists. Consequently, analysis in adapted spaces actually concerns regularity results, once existence of solutions in Morrey spaces is established. As we remarked in the Introduction, choosing BM spaces, which are of Besov- type, over Morrey spaces has additional advantages. First of all, there is a better pseudo-differential and para-differential calculus, especially with respect to point- wise multiplication. Secondly, BM spaces of negative index contain distributions 0 more singular than Radon measures. For example, p.v.(1/x) 1,1, (R), so that ∞ 0 n n/p 1 n ∈N (0,...,0, p.v.(1/x1)) n,1, (R ) p,p/n,− (R )aslongasp>n(see [KY], p. ∈N ∞ ⊂N ∞ 969). Note that Theorem 5.12 below applies with a(x)=δ (0,...,0, p.v.(1/x1)) as initial data, provided δ is sufficiently small. As mentioned in Section 2.2, Koch and Tataru have proved existence and unique- 1 ness of global solutions with data small in BMO− . Their result does not include n/p 1 Theorem 5.12 entirely, as one can take data small in p,1, − ,aslongasp>n,and n/p 1 1 N ∞ p,1, − * BMO− (see also [CX]). N In∞ [KY], Kozono and Yamazaki focus on scalar perturbation of the heat equa- tions, and the Navier-Stokes equation. However, at least as far as existence and uniqueness are concerned, the proof applies to systems of equations, as long as the non-linear part satisfies certain homogeneity constraints. We first recall Kozono and Yamazaki’s results and then discuss some generaliza- tions.

13For an overview of recent results we refer to [Ya].

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5.2. Semi-linear heat equations. We consider the Initial Value Problem: ∂ u(t, x)=∆u(t, x)+f(u(t, x)) , (5.5) t ( u(0,x)=a(x) , where u :(0, + ) Rn C and f : C C. We assume that f is a locally Lipschitz function∞ satisfying× → → γ 1 γ 1 (5.6) f(z) f(w) C z w (1 + z − + w − ) , | − |≤ | − | | | | | for some γ>1. Equations of this form arise, for example, in mathematical biology and control theory [Wu]. One of the main results in [KY] is the following theorem. Theorem 5.4 ([KY]). Suppose that f satisfies (5.6) and assume p, q, s are numbers such that γ q p, ≤ ≤ n(γ 1) < 2p, (5.7)  −  2/γ < s < 0 ,  − s n/p 2/(γ 1) . ≥ − −  s Then, there exist δ, K > 0 such that for every a Np,q, satisfying ∈ ∞ js p (5.8) lim sup 2 ψj(D)a Mq <δ, j k k →∞ there are T>0 and a mild solution u(t, x) of (5.5) on [0,T) Rn such that × s/2 p (1) sup0Lorentz space L − ∞ for the range of indices above, this extra hypothesis cannot be eliminated (see [KY], pp. 991–992). However, it is redundant for s> n/p 2/(γ 1), as we will show later. − − Remark 5.5. (5.8) is also the analogue of the condition a M◦n in Theorem 4.3 of ∈ 1 ◦n [Tay3], where M1 is the set of functions u for which n/p n/q νpq(u) = lim sup R − u Lq(B(0,R)) =0. R 0 k k → Note further that, if u(t, x) is a solution to the problem (5.5), then, for each 2/(γ 1) 2 positive λ, λ − u(λ t, λx) is also a solution. So, the norm of the homogeneous n/p 2/(γ 1) space p,q, − − is exactly the one invariant under such scaling, that is, this space isN a limit∞ space and n/p 2/(γ 1) is the critical index. Accordingly, one − −

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does not expect a simple contraction mapping argument to hold and the Picard iteration must be inspected directly. We sketch the proof of the theorem, since similar arguments will be used later. The main tool consists of the following estimates on the heat semigroup. Lemma 5.6 ([KY]). Let ` s and 1 q p< .Then ≥ ≤ ≤ ∞ t∆ (s `)/2 (5.9a) e u ` C (1 + t ) u s , Np,q, − Np,q, k k ∞ ≤ k k ∞ t∆ (s `)/2 (5.9b) e u ` Ct − u s . p,q, p,q, k kN ∞ ≤ k kN ∞ If `>s,thenalso t∆ (s `)/2 (5.10a) e u ` C (1 + t − ) u s , Np,q,1 Np,q, k k ≤ k k ∞ t∆ (s `)/2 (5.10b) e u ` Ct − u s . p,q,1 p,q, k kN ≤ k kN ∞ The lemma is an easy consequence of the results of Section 3. Lemma 5.7 ([KY]). Suppose that 1 q p< and `>s. Then there exists a ≤ ≤ ∞ s positive constant A = A(p, q, s, `) such that, for every u Np,q, and every B such that ∈ ∞

js p (5.11) A lim sup 2 ψj(D)u Mq 0 for which (` s)/2 t∆ (5.12) sup t − e u N ` Bochner integral. Likewise, the initial value is attained only in a weak sense; how weak is precisely the content of Part 2 in Theorem 5.4. Picard iterates are defined, as usual, by t∆ u0(t, x)=e a(x) , t t∆ (t τ)∆ u1(t, x)=e a(x)+ e − f(u0(τ,x)) dτ , Z0 (5.14) . . . . t t∆ (t τ)∆ uj+1(t, x)=e a(x)+ e − f(uj(τ,x)) dτ. Z0 Let also vj = uj uj 1. Next, we obtain− bounds− on the non-linear part of (5.13), using (5.6) and inclusion relations for Morrey spaces under pointwise multiplication. 0 p Since Np,q,1 Mq , in view of Lemma 5.7 we introduce an auxiliary function space. ⊂

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n Definition 5.8. Let = T be the space of measurable functions u on (0,T) R such that A A ×

s/2 p (5.15) u =supt− u(t, ) Mq < . k kA 0 0, independent of a,andatimeT 1 such that ≤ (5.16) A0

(2) Moreover, there exists C1 > 0 independent of T such that, j 0, ∀ ≥ γ sγ/2 (5.17) Aj+1 A0 + C1(A + T − ) . ≤ j

0 p Part 5.9 follows immediately from Lemma 5.7 and the inclusion Np,q,1 Mq . Part 5.9 is, instead, a consequence of Lemma 5.6 and the Lipschitz condition⊂ (5.6) n(γ 1)/p 0 p on f by means of the inclusion N − N M . The details of the proof p/γ,q/γ,1 ⊂ p,q,1 ⊂ q are in [KY]. The lemma implies, in particular, that the sequence Aj is bounded. As a γ { } γ matter of fact, for T small enough, Aj+1 C0δ>A0.Furthermore,αδ ωδ = ≤ 2C0γδ/(γ 1), which gives Aj ωδ for all j. − ≤ An estimate similar to (5.17) can be obtained for Bj, using again (5.6) and the bound on Aj.

Lemma 5.10 ([KY]). There exists C2 independent of T 1 such that the inequality ≤ γ 1 s(1 γ)/2 (5.18) Bj+1 C2Bj(ω(δ) − + T − ) ≤ holds j 0. ∀ ≥ At this point, convergence of the series j Bj follows easily for δ and T suf- ficiently small, which is in turn enough to show uj is a Cauchy sequence in P { } . The limit u is clearly a solution to (5.13). Additionally, Aj u so that A →k kA u ωδ = K. Part (1) of Theorem 5.4 is now established. k kPartA ≤ (2) of Theorem 5.4 is a consequence of the inclusion p 0 n/p (5.19) Mq Np,q, B− , . ⊂ ∞ ⊂ ∞ ∞ In a fashion similar to the proof of Lemma 5.9, it ensues that t (t τ)∆ s n/p u v = e − f(u(τ, )) dτ C([0,T),B −, ) , ∈A⇒ · ∈ ∞ ∞ Z0 s n/p n/p s where B −, is given the weak- topology as dual of B1,1 − . As pointed out by ∞ ∞ ∗ Cannone [Can], the solution is composed of a trend w = et∆a and a fluctuation v, which is more oscillating and better behaved for t 0+. Part (3) of Theorem 5.4 is an immediate consequence→ of Lemmas 5.7 and 5.9, while part (4) is a standard regularity argument. We refer to [KY] for details.

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It remains to prove uniqueness.

Proposition 5.11 (KY). Let f, p, q, s, γ satisfy the hypotheses of Theorem 5.4. Then, for every T (0, ] and every a 0, there is at most one solution u of ∈ ∞ ∈S (5.5) on (0,T) Rn such that × s (1) u L∞((0,T0),Np,q, ) T 0 if T 0

s When s = n/p 2/(γ 1), as emphasized earlier, the norm of p,q, is invariant under rescaling. Therefore,− − one would expect global-in-time existenceN ∞ to hold for this limit case, in particular, in the form of self-similar solutions [KY2]. However, in order to exploit the action of time dilations, a homogeneity constraint needs to be imposed on f. Assume, then, that f satisfies

γ 1 γ 1 (5.21) f(z) f(w) C z w ( z − + w − ) , | − |≤ | − | | | | | and f(0) = 0, which, in particular, implies (5.22) f(z) C z γ . | |≤ | | This choice gives bounds similar to (5.17) and (5.18) that are actually independent of T . In addition, (5.7) becomes

γ>1+2/n , (5.23) γ q p,  ≤ ≤  n(γ 1) < 2p

s/2 p (5.24) u T =supt− u(t, ) q < . k kA t>0 k · kM ∞

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We have the analogs of Lemma 5.9 and Lemma 5.10:

(5.25a) A0 C3δ, ≤ γ (5.25b) Aj+1 A0 + C4A , ≤ j γ 1 (5.25c) Bj+1 C5ω(δ) − Bj , ≤ with ω(δ)=C3γδ/(γ 1) for δ [0,δ0]. As before, (5.25a) and (5.25b) give − ∈ uniform bounds on the Aj, while (5.25c) implies convergence of the series j Bj. Uniqueness then follows from Proposition 5.11. P 5.3. The Navier-Stokes equation. We consider, now, an Initial Value Problem for the Navier-Stokes system on Rn:

∂tu(t, x)=∆u(t, x) u u(t, x) p(t, x)+f(t, x) , − ·∇ −∇ (5.26)  div u(t, x)=0,  u(0,x)=a(x).

This is a system of n+1 equations in the unknown u (the velocity vector field)andp (the pressure scalar field). The initial data a is assumed to be divergence-free and f represents body forces. The first equation is basically conservation of momentum, or Newton’s second law, and the second, the continuity equation, is conservation of mass.14 The density ρ and the viscosity coefficient ν do not appear in the equations, since they are constant and are conventionally set equal to one. As a matter of fact, the methods employed here rely heavily on the smoothing action on the heat semigroup and they are valid only for strictly positive ν. In order to rewrite (5.26) in integral form, we are going to make some further (mild) assumptions. First of all, if the forces are conservative and admit a potential g, f = g can be absorbed in the pressure term. Secondly, from a physical point of view,∇ the role of the pressure is to enforce incompressibility. Mathematically, the Hodge decomposition ensures that gradients are orthogonal to divergence-free vector fields. Thus P( p)=0,whereP is the so-called Leray projection onto the space of divergence-free∇ vector fields, at least when p is sufficiently regular. Finally, note that u u =div(u u)ifdivu = 0. This last expression is more appropriate when dealing·∇ with distributions.⊗ Therefore, (5.26) reduces to

∂tu(t, x)=∆u(t, x) P div(u u)(t, x) , (5.27) − ⊗ ( u(0,x)=a(x) . The original system is now in the form of a single (vector) semi-linear parabolic equation. The non-linear part of (5.27) is the product of a first-order (non-local) operator P , and a bilinear form : Q (5.28) P div(u u)=P (u, u) . ⊗ Q In particular, it has a scaling factor γ = 3 (in the notation of the previous section); s consequently, one expects to have local-in-time solutions for initial data in Np,q, , ∞

14For a derivation of these equations we refer to [ChM].

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n/p 1 s n/p 1, and global-in-time solutions for initial data in p,q, − . Kozono and Yamazaki≥ − consider the limit case in [KY]. N ∞ Theorem 5.12 ([KY]). Let 1 q p< and p>n.Then,thereexistδ, K > 0 n/p ≤1 ≤ ∞ such that, for every a Np,q, − satisfying ∈ ∞ j(n/p 1) p lim supj 2 − ψj(D)a Mq <δ, (5.29) →∞ k k (div a(x)=0, there are T>0 and a mild solution u(t, x) of (5.27) on [0,T) Rn such that 1/2 n/4p × (1) sup0

n Definition 5.14. Let = T be the space of measurable functions u on (0,T) R such that G G × 1/2 n/4p (5.33) u =supt − u(t, ) M 2p < . k kG 0n,α=1 n/q . k k∗ t>0{ k k } −

Now, let uj be the Picard iterates and set Gj = uj , Fj = vj ,with k kG k kG vj = uj uj 1, as before. Lemmas− similar− to Lemma 5.9 and Lemma 5.10 hold in this case as a consequence of Lemmas 5.6, 5.7 and 5.13.

Lemma 5.15 ([KY]). (1) Under the hypotheses of Theorem 5.12, there exists C0 > 0, independent of a,andatimeT 1 such that ≤ (5.35) G0

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(2) Moreover, there exists C1 > 0 independent of T such that, for all j 0, ≥ 2 (5.36) Gj+1 G0 + C1 G . ≤ j

The lemma gives a uniform bound on Gj,namelyGj ω(δ), j 0, with 2 ≤ ∀ ≥ ω(δ)=2C0δ/(1 + √1 4C0C1δ) the smaller root of C1x x + C0δ =0. − − Lemma 5.16. There exists C2 independent of T 1 such that the inequality ≤ (5.37) Fj+1 C2ω(δ) Fj ≤ holds j N. ∀ ∈ Now, choose δ so small that ω(δ) 1/2C2. Then, Fj Fj/2sothat Fj < ; ≤ ≤ j ∞ that is, the sequence uj converges in to a solution of (5.30), with u ω(δ)=K, and Part (1){ is} proved. The restG of the theorem follows as TheoremP k kG 5.4.≤ Uniqueness is established via a simple modification of Proposition 5.11. Finally, we have global-in-time existence if a sufficiently small initial data a n/p 1 ∈ p,q, − is selected parallel to the analysis of (5.5). N ∞ 5.4. Extensions. As we discussed earlier, it is rather natural to expect (local) existence to hold in the subcritical range of indices for arbitrary initial data. Recall that

s s/2 t∆ p (5.38a) Np,q, = f 0 sup t− e f Mq < , ∞ ∈S | 0

s s/2 t∆ p (5.38b) p,q, = f 0 sup t− e f q < , N ∞ ∈S | 0n/p 2/(γ 1) . − − s  Then, for all a Np,q, ,thereexists T>0 and a unique mild solution u to (5.5) n ∈ ∞ on [0,T) Rx that enjoys all the properties of Theorem 5.4. × Proof. We prove existence and uniqueness only. Let ,Aj ,Bj be as in the proof A of Theorem 5.4. Show first that the sequence Aj is bounded for sufficiently small { } s T . By (5.38a), if T 1, A0 u0 Np,q, . As in the proof of Lemma 5.9 ([KY], pp. ≤ ≤k k ∞ 993-994), we obtain the following estimate: s/2  γ sγ/2 (5.40) uj+1(t) u0(t) p C1 t t (A + T − ) , k − kMq ≤ j with  =1+(γ 1)(s/2 n/2p). Note that >0, because s>n/p 2/(γ 1). Now, choose −T so small− that − −  sγ/2 γ (5.41) C1 T (A0 + T − ) 1/2 ≤

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 γ 1 and proceed by induction. Assume that C1 T Aj − 1/2 j k.ThenAj+1  sγ/2 ≤ ∀ ≤ ≤ 1/2 Aj + ν with ν = T − small. In fact,  sγ/2 > 0, as 2p>(γ 1)n. Consequently, − −

1 i (5.42) Aj+1 A0 + ν 2− 2(A0 + ν) . ≤ 2j+1 ≤ 0 i j ≤X≤ In particular, by means of estimate (5.41), for j = k,

 γ 1  γ 1 γ 1 1 γ 1 1 C1 T A − C1 T 2 − (A0 + ν) − 2 − = , k+1 ≤ ≤ 2γ 2 which proves the induction step and gives finally

(5.43) Ak 2(A0 + ν) K, k. ≤ ≡ ∀ We show next that the series Bj converges. This is enough to prove uj u in j → . Following the proof of Lemma 5.10 ([KY], pp. 995-996), we obtain the estimate: A P s/2  γ 1 s(1 γ)/2 (5.44) vj+1 p C2 t t Bj [2A0 +2ν] − + T − , k kMq ≤   with  and ν as before. Again, choose T1 T so small that ≤  γ 1 s(1 γ)/2 1 (5.45) C2 T [2A0 +2ν] − + T − . 1 ≤ 2   Then, Bj+1 1/2Bj,thatis, j Bj < . As far as≤ uniqueness is concerned, we∞ cannot directly quote Proposition 5.11, because K may not be small forP T small. But here (5.20) is replaced by

 γ 1 s(1 γ)/2 H(T 0) C2(T 0) K − + T − H(T 0). ≤ Hence H(T ) 0forT sufficiently small.  ≡  Remark 5.18. The constants C1 and C2, and thus the length of the existence time T , depend only on the size of the initial data. Therefore, the solution persists for

s as long as u(t) Np,q, stays finite. k k ∞ A similar derivation, using instead Lemmas 5.15 and 5.16, yields the following conclusion for the Navier-Stokes equation. Proposition 5.19. Let p, q, s be numbers such that 1 q p, ≤ ≤ (5.46) p>n, n/p 1 0 and a unique mild solution u to (5.27) n ∈ ∞ on [0,T) Rx that enjoys all the properties of Theorem 5.12. × Note that the condition s<0 is used in deriving (5.40), which requires s(1 γ) > 0, and here γ>1. − To improve this local existence result we turn to the theory of adapted spaces [Can], which we briefly recall. These are functional Banach spaces E (with transla- tion-invariant norm) such that

(5.47) ψj(D)(fg) E ηj f E g E , j Z , k k ≤ k k k k ∀ ∈

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where

j (5.48) 2−| | ηj < . ∞ j X The two conditions above are sufficient (but clearly not necessary) to give a local Lipschitz condition on the map N, so that local-in-time existence follows. Intu- itively, it is enough to bound the non-linear part of N. When frequencies are re- t∆ j j stricted to the shell Dj, the operator e P has symbol comparable to 2 exp( t 4 ). So, if t>4j, the negative exponential will control the growth of σ(P ) and the− qua- dratic part. If t 4j, the smoothing effect of the heat semigroup is small, but ≤ j 1 Pψj(D)(u u) E then contributes at most 4 n/p,andfors n/p if also r = 1. Para-differential calculus will give further estimates. ≥ In fact, if n/2p

j(n/p s) ψj(D)(uv) N s C 2 − u v 2s n/p (5.49) k k p,q,r ≤ k · kNp,q,r− j(n/p s) C 2 − u N s v N s , ≤ k k p,q,r k k p,q,r j` ` since ψj(D)isO(2 )inOPS1−,0. Therefore,

j(n/p s) (5.50) ηj 2 − ∼ and (5.48) holds, because p>n.Ifs = n/p, we must use (3.69) instead to conclude

j (5.51) ηj 2 , >0, . ∀ which is still sufficient for (5.48). We cannot expect to reach s = 0 this way, at least for small values of q, because when q<2, fg may not even be locally integrable. We assume, therefore, q 2 and we look more carefully at the paraproduct decomposition. As shown in [Can],≥

ψj(D)(fg)=ψj(D)fφj 2(D)g + ψj(D)gφj 2(D)f − − (5.52) + ψj(D) ψk(D)fψk(D)g ,   k j X≥   where φj = ψk. Furthermore, since ψjψk 0for j k > 1, it is enough to k j ≡ | − | bound the Morrey≤ norm of each of the terms in (5.52). We startP by recalling that, if f L1 ,then ∈ loc p t∆ n/2p (5.53) f M e f Ct− , ∈ 1 ⇔ | |≤ t∆ j for 0

jn/p (5.54) φj(D)f L C 2 φj(D)f M p . k k ∞ ≤ k k 1

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p H¨older’s inequality then gives the same estimate for Mq , and Lemma 3.16 allows us to conclude that

φj(D)fψj (D)g p φj(D)f L ψj(D)g p k kMq ≤k k ∞ k kMq (5.55) jn/p 2js C 2 2− f N s g N s , ≤ k k p,q,r k k p,q,r if also s>0. As for the third term in (5.52), for notational convenience we set ψk(D)f = fk, ψk(D)g = gk. We observe first that

(5.56) f g p/2 f p g p f p g p , k k M k Mq k M k Mq k Mq k k 1 ≤k k k k q0 ≤k k k k

where q0 is the conjugate exponent of q and q q0,sinceq 2. ≥ p/2 ≥p n/p While it is not true, in general, that p(D):M M ,ifp S− [Tay3], 1 → q ∈ 1,0 p/2 p ψj(D)doesmapM1 into Mq . However, proceeding this way yields an estimate for ηj that is not sharp (cf. (5.54)). Instead an elementary calculation shows that

1/q0 1/q p h h h p/q . Mq L∞ k k ≤k k k kM1

Here, h = ψj(D)(fkgk). Hence we use (5.54) again for the first piece with p/q in place of p:

1/q0+1/q p jnq/p 1/q0 (5.57) ψj(D)(fkgk) Mq C (2 ) ψj(D)(fkgk) p/q . k k ≤ k kM1 p/q p/2 We then need an estimate of the norm in M1 in terms of M1 for functions of j the form ψj(D)u. Because ψj is supported on ξ 2 , we can restrict t in (5.53) j | |∼ to 0

p/2 t∆ nq/2p jn(q/p 2/p) ψj(D)u M e ψj(D)u Ct 2 − , ∈ 1 ⇒ | |≤ as q/p 2/p 0, which is equivalent to − ≥ jn(2/p q/p) (5.58) ψj(D)u M p/q C 2 − ψj(D)u M p/2 . k k 1 ≤ k k 1 Combining (5.56), (5.57), and (5.58), gives

jn(q 1)/p jn(2/p q/p) ψ (D)(f g ) p C 2 2 f g p/2 j k k Mq − − k k M k k ≤ k k 1 jn/p C 2 fk p gk p . ≤ k kMq k kMq Finally, if s>0, similarly to (5.55),

2sk jn/p 2js (5.59) ψj(D)( fkgk) p C ( 2− )2 2− f N s g N s . k kMq ≤ k k p,q,r k k p,q,r k j k 0 X≥ X≥ Therefore, as before, j(n/p s) ηj 2 − . ∼ We gather these results in the following corollary.

s Corollary 5.20. The BM space Np,q,r is adapted to the Navier-Stokes equation for (1) s>n/2p,ifp>n, 1 q<2, r [1, ], (2) s>0,ifp>n, 2 q≤ p< , r∈ [1∞, ]. ≤ ≤ ∞ ∈ ∞

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In view of the Sobolev-type embedding of Theorem 2.4, the solution given by the previous corollary will behave like a classical solution for large s,verymuchin the spirit of Sobolev spaces. Let us summarize our extensions so far. For the inhomogeneous BM space s Np,q, , we have local-in-time existence and uniqueness if ∞ n/p 1 n/2p, − for arbitrary initial data. If s = n/p 1, existence and uniqueness hold for data satisfying (5.29). − n/p 1 If s = n/p 1, existence is global in time in the homogeneous BM space p,q, − for sufficiently− small initial data. N ∞ We conclude this section by comparing the previous results with those of Karch. In [Kr], he considered an IVP with data in BEα for a similar class of parabolic equations; namely, u =∆u + B(u, u) , (5.60) t (u(0) = a. Here, B is a bilinear form with scaling order b<2.

Definition 5.21. B is said to have scaling order equal to b R if ∈ b B(uλ,vλ)=λ (B(u, v))λ ,

where uλ(x)=u(λx). Note that the IVP (5.5) is of the form (5.60) (with b =0)onlyiff is homogeneous of degree 2 (so, γ = 2), while NS becomes (5.60) for b = 1. Recall, once again, α s p that BEp can be identified with p,q, if Ep = q and α = s, s<0. Then the existence criteria established in SectionN ∞ 5.2 (TheoremM 1 in [KY])− and in Section 5.3 (Theorem 3 in [KY]) become exactly the content of Proposition 5.1 and Theorem 5.1 in [Kr]. At the same time, other results in [Kr] can now be applied to the BM spaces, for example persistence and regularity of solutions. We just mention the following asymptotic condition.

Theorem 5.22 ([Kr]). Let u1 and u2 be two (global) mild solutions to the Navier- n/p 1 Stokes equation (5.27) with sufficiently small initial data a1, a2 p,q, − respec- tively. If ∈N ∞

1/2 n/4p t∆ p (5.61) lim t − e (a1 a2) q =0, t + M → ∞ k − k then

1/2 n/4p p (5.62) lim t − u1(t) u2(t) q =0. t + M → ∞ k − k n/p 1 n/p 1  Condition (5.61) is satisfied, for example, if aj p,q, − p,q, − − , >0, j =1, 2, given again the identification (5.38). ∈N ∞ ∩N ∞

5.5. Analysis on compact manifolds. Let us consider again the Navier-Stokes and semi-linear heat equations now with u(t, x) a function on [0, + ) M,and s ∞ × a Np,q, (M). As usual, M stands for a smooth, compact, Riemannian manifold without∈ boundary.∞

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The analysis here is complicated by curvature terms and the non-local nature of the equation we consider. On the other hand, we do not have to deal with the behavior at infinity. Therefore, we will extend only the local existence results. We start by considering again semi-linear heat equations. Since we cannot exploit finite propagation speed and work in local coordinates, even for small t, we will need to use the intrinsic characterization of BM spaces obtained in Section 4.2. The idea is to “mimic” the proof of Theorem 5.4 on Euclidean space by replacing ψj(D)withψj(√ ∆), ∆ the Laplace operator on M. − Theorem 5.23. Suppose that f satisfies (5.6) and assume p, q, s are numbers such that γ q p, ≤ ≤ n(γ 1) < 2p, (5.63)  −  2/γ < s < 0 , − s n/p 2/(γ 1) .  ≥ − −  s Then, there exist positive constants δ, K, such that, for every a Np,q, (M) satisfying ∈ ∞

js √ p (5.64) lim sup 2 ψj( ∆)a Mq (M) <δ, j k − k →∞ there are T>0 andauniquemildsolutionu(t, x) to (5.5) on [0,T) M such that ×

s/2 p (5.65) sup t− u(t, ) Mq (M) K. 0

s1 s2 s (5.67) (N (M),N (M))θ,r = N (M) , (1 θ)s1 + θs2 = s, p,q,r1 p,q,r2 p,q,r − which, in turn, is easily obtained from (4.53) exactly as in the Euclidean case (cf. [KY], pp. 986-987). Next, we prove a version of Lemma 5.7. Lemma 5.24. Let 1 0 such that

(` s)/2 t∆ (5.69) sup t − e u N `

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Proof. The proof is similar to that of Lemma 5.7, but again15 we have to treat an infinite asymptotic expansion for the symbol of ψj(√ ∆). − By the analogue of (5.10b) obtained above, there exists C0 such that

t∆ (s `)/2 (5.70) e u ` C t − u s , Np,q,1 0 Np,q, k k ≤ k k ∞

with C0 independent of T ,0m+ N,  P and  m (1 ψ0(2− ξ x))ψj( ξ x) − | | | | 0 j

ψj(√ ∆)Φm(√ ∆) = Rj∞ , if j>m N, (5.74) − − − ψj(√ ∆)(Id Φm(√ ∆)) = R˜∞ , if j

js s √ √ p u1 Np,q, C1 sup 2 ψj( ∆) Φm( ∆)u Mq , k k ∞ ≤ j m+N{ k − − k } (5.75) ≤ js s √ √ p u2 Np,q, C2 sup 2 ψj( ∆) (Id Φm( ∆))u Mq , k k ∞ ≤ j m N{ k − − − k } ≥ − for some positive constants Cj, j =1, 2. Moreover, because of (5.73a) and (5.73b), ˇ these constants depend only on ψj L1 . It follows from (5.71) that k k

js s √ p (5.76) u2 Np,q, K sup 2 ψj( ∆)u Mq Kα, k k ∞ ≤ j m N{ k − k }≤ ≥ −

15Cf. Section 4.2.

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with K depending only on ψj .ChooseA = C0 K. Then, for 0

s s/2 t∆ p (5.79) u Np,q, (M) sup t− e u Mq (M) < . ∈ ∞ ⇔ 01,andletγ,p,q,s be numbers such that γ q p, ≤ ≤ n(γ 1) < 2p, (5.80)  −  2/γ < s < 0 , − s>n/p 2/(γ 1) . − − s  Then, for all a Np,q, (M),thereexistsT>0 and a unique mild solution u to (5.5) on [0,T) ∈M. ∞ × We now turn to the Navier-Stokes system. As we data mentioned at the be- ginning of Section 5, curvature adds an extra term to the momentum equation in (5.26). The “correct” version of the Navier-Stokes equations on a curved manifold is the following system [EM]:

∂tu(t, x)=∆u(t, x)+2P Ric u(t, x) P div(u u)(t, x) , − ⊗ (5.81)  div u(t, x)=0,  u(0,x)=a(x) , jk jik where ∆ is the Laplace-Beltrami operator and Ric = Ri is the Ricci tensor on M, i.e., the contraction of the curvature tensor Rijkl. Without going into detail (see [Tay3] and the discussion there), we just say that the Laplacian enters the momentum equation by applying the divergence theorem to the stress or deformation tensor S, which represents the forces exerted on some volume of fluid across a unit surface element. So, one should actually have div S in place of the Laplacian in the equation. But on Rn div Su =∆u, while on a manifold div Su=∆u +2Ricu,ifdivu =0.

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Here, we “mimic” the proof of Theorem 5.12. The term P Ric u does not sub- stantially modify the proof, because it is an operator of order 0. Proposition 5.26. Let 1 n.Then,thereexistδ, K > 0 n/p 1 ≤ ∞ such that for every a Np,q, − (M) satisfying ∈ ∞ j(n/p 1) √ p lim supj 2 − ψj( ∆)a Mq (M) <δ, (5.82) →∞ k − k (div a(x)=0, there are T>0 and a mild solution u(t, x) of (5.81) on [0,T) Rn such that × 1/2 n/4p (5.83) sup t − u(t, ) M 2p(M) K. 0

(5.87) ψj(√ ∆)fg E ηj f E g E, k − k ≤ k k k k where ηj still satisfies (5.48). To prove that the map N is a contraction for short-time, we utilize the asymptotic expansion for the heat parametrix in local coordinates (see [Tay1], Vol. 2), and arguments similar to those leading to Lemma 5.24.

Acknowledgements The author thanks her Ph.D. adviser, Michael E. Taylor, for his continuous support and encouragement, and Richard Beals, Roberto Camassa, Peter Jones, Norberto Kerzman, and Mark Williams for helpful discussions.

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Department of Mathematics, Yale University, P.O. Box 208283, 10 Hillhouse Ave., New Haven, Connecticut 06520-8283 E-mail address: [email protected]

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