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Home , Interpolation space, Sobolev space

Characterizations of Screened Sobolev Spaces Noah Stevenson and Ian Tice DEPARTMENTOF MATHEMATICAL SCIENCES,CARNEGIE MELLON UNIVERSITY

An approach which is promising in the resolution of this issue is to swap the inhomogeneous Sobolev spaces for their homogeneous counterparts, Does the topology on the screened Sobolev spaces Background W˙ k,p (Ω). The upshot is that the latter families allow for a larger variety of Research admit a Fourier characterization? behaviors at infinity. For this approach to have any chance of being fruitful in the Leoni and Tice gave the following partial frequency & Motivation study of boundary value problems, it is essential to identify and understand the Questions n characterization of these spaces: for f ∈ S R it holds Partial differential equations (PDEs) are trace spaces associated to the homogeneous Sobolev spaces. Recall that the trace Z 1  2s 2 ˆ 2  2 central to the modeling of natural space associated to a on a domain is a space of functions holding Can the screened Sobolev spaces be [f]W˜ s,2  min{|ξ| , |ξ| } f (ξ) dξ , (2) (σ) n understood through interpolation theory? R phenomena. As a result, the study of these the possible boundary values and, for higher order Sobolev spaces, powers of the where 0 < s < 1 and constant screening function σ = 1. equations and their solutions has held a normal . Interpolation theory is a set of tools which This expression suggests that the high-mode part of a prominent place in for For functions of finite energy in planar strips these trace spaces were first s,2 take certain pairs of topological vector member of W˜ behaves like a member of the centuries. Rarely do PDE admit explicit identified by Strichartz in [Str16]. He found that for functions f defined on spaces, called the endpoint spaces, and (1) solutions and more round-about techniques R 2 Sobolev-Slobodeckij space Hs while the low mode part is U = R × (0, 1) satisfying U |∇f| < ∞ the fractional regularity of the return a space which is ‘in between’ the two. are necessary to gain deeper trace functions is measured according to the : On bounded domains with Lipschitz allowed to grow and behave like an element of H˙ 1. The Depicted above is a portion of the graph of understanding. This catalyzes the x+1/2 2 1 boundaries, the trace spaces associated to issue with this result is that not every member of the screened  Z Z |f (x) − f (y)|  2 of p-finite energy on an infinite strip-like domain. development of novel analytical tools and [f] 1 = dy dx inhomogeneous Sobolev spaces are Sobolev spaces is Schwartz and no such result previously H˜ 2 2 The trace of this function is highlighted in red. spaces of functions. R x−1/2 |x − y| members of the scale of Besov spaces. The existed on the density of the Schwartz class. Hence, the where s = 1 − 1/p and σ : n → (0, ∞] is a lower semicontinuous screening Perhaps the most famous functions Strichartz’s results were subsequently expanded upon by Leoni and Tice in [LT19]. R latter family, it turns out, is characterized as above heuristics on high and low mode behavior may or may function, whose definition is dependent on η . The space of locally integrable functions on spaces to emerge from this development They considered infinite strip-like domains of the form ± ‘in between’ a Lebesgue space and an not apply to every function in a screened Sobolev space. 0 n+1 0 0 n+1 are the classical inhomogeneous Sobolev U = {x = (x , xn+1) ∈ R | η−(x ) < xn+1 < η+(x )} ⊂ R , f for which the seminorm (1) is finite is called the screened Sobolev space with parameters inhomogeneous Sobolev space by the real The starting point for a more robust Fourier characterization n ˜ s,p n method of interpolation. Thus, it is natural to k,p where η− < η+ are Lipschitz on R and functions defined on these domains of 0 < s < 1, 1 ≤ p < ∞, and σ. It is denoted W R . The moniker ‘screened’ is is to first ask if the screened Sobolev spaces are naturally spaces, W (Ω). These are (σ) expect that a similar characterization holds p-finite energy: e.g. R |∇f|p < ∞, 1 ≤ p < ∞. In this more general included within the space of objects for which it makes sense indispensable when studying PDE in U justified by the fact that this seminorm screens difference quotients and only allows certain for the screened Sobolev spaces. An to take the - the tempered distributions. If case, the fractional regularity of the boundary values is quantified via: bounded domains; however, as soon as one ones to affect the seminorm’s value. affirmative answer to this question with this is true, then the expression could have a suitable Z Z p 1 moves to unbounded domains, these  |f (x) − f (y)|  p As a result of their groundwork development of these screened spaces, Leoni and Tice classical function spaces as endpoint ˜ s,2 [f]W˜ s,p = n+sp dy dx , (1) interpretation for all f ∈ W . In the cases p 6= 2 a inhomogeneous Sobolev spaces may fail in (σ) n were able to provide necessary and sufficient conditions for the solvability of certain spaces will give a clear and quantified (1) R B(x,σ(x)) |x − y| providing a suitable functional setting for the quasilinear elliptic boundary value problems in the strip-like domains. The screened Sobolev connection between the screened spaces Littlewood Paley Decomposition of these spaces could reveal problem at hand. spaces have also been useful in recent studies of the Muskat problem [Ngu19,FN20,NP20]. and more well-studied function spaces. the frequency behavior.

F (A) and F (B) satisfy the Depending on one’s choice of function spaces, Variant on Peetre’s method: f = ∆−4f + ∆0f + ∆4f Littlewood-Paley characterizations interpolation property. The upshot, for y1 Littlewood-Paley offers suitable replacements for truncated interpolation TOOL #1: our purpose, is that when this TOOL #2: (3) to reveal what orthogonality relations hold. The Lebesgue space Lp n for 0, t−1K (t, x) R construction is explicit it will be clear Peetre’s real method does not quite 1 < p < ∞ has the following Interpolation which properties the space F (A) capture the screened Sobolev spaces Littlewood-Paley Littlewood-Paley decomposition frequency space characterization: The aim of interpolation theory can be inherits from the couple [Tri78]. Here is Γ(x) as interpolation spaces. Tice and I The Littlewood-Paley theory concerns  1/2 This decomposition is the first step in P 2 developed the following asymmetric kfkLp  j∈ |∆jf| . described as follows. A pair of seminormed a showcase two of these constructions the manifestation of the Fourier our search for orthogonality. It allows us Z Lp + spaces X for i ∈ {0, 1} are called a relevant to the analysis of screened variant. For σ ∈ and s, q as transform’s orthogonality properties on f i R to write a given tempered distribution , The homogeneous Sobolev space compatible interpolation couple if there is Sobolev spaces. before, we defined the trunated spaces other than the familiar L2, like Lp i.e. , as an infinite ∆−4f k,p n a X such that X ⊆ X . W˙ for 1 < p < ∞ and i Peetre’s real method of interpolation space via for p 6= 2,Holder¨ spaces, and Sobolev series over the integers of tempered R (σ) n (σ) o k ∈ Given another interpolation couple Yi for (X ,X ) = x ∈ X + X | [x] < ∞ spaces. Deeper understanding of this distributions in which each of the terms N has the characterization: interpolation [Pee68] 0 1 s,q 0 1 s,q 1/2 i ∈ {0, 1} belonging to Y, one asks for k P jk 2 Z 1/q orthogonality leads to a refined study of act ‘independently’. The jth term, ∆0f ∇ f  2 |∆jf| . y0  dt Lp j∈ seminormed spaces X ⊆ X and Let (X , [·] ) for i ∈ {0, 1} be (K (t, x) , 0) (σ) −s q ∆ f Z Lp i i [x]s,q = t K (t, x) . in function spaces. The extent denoted j , is roughly the Y ⊆ Y satisfying the interpolation an interpolation couple. Their algebraic (0,σ) t to which the Littlewood-Paley theory f ∆ f The homogeneous localization of at spatial frequencies 4 s,p property: whenever T : X → Y is linear sum is equipped with the following The Gagliardo diagram above shows a Because this seminorm only cares characterizes spaces of functions is j B˙ n for 1 < p < ∞, geometric interpretation of the K function. |ξ| ≈ 2 . More explicitly, there is a q R and the restriction to Xi is continuous 1-parameter family of : about the small t behavior, a special remarkable. The above depicts the 1 ≤ q ≤ ∞, and 0 < s < 1 has For some x ∈ X + X we set special smooth function ψ supported in X → Y for i ∈ {0, 1} it then holds 0 1 role is inherited by the second factor, Littlewood-Paley decomposition i i K (t, x) = inf [x0]0+t [x1]1 . Orthogonality of the Fourier transform on the characterization: x=x0+x1  2 the annulus 1/2 ≤ |ξ| ≤ 2 with of a limited function f into that TX ⊆ Y and the restriction to X is Γ(x) = y ∈ R | ∃x =. x0 + x1, [xi]i ≤ yi X1. In fact, we proved the following 2 −j  n sj L is easy to understand. If {f } is a P n  t ∈ (0, ∞) , x ∈ X0+X1. j j∈ ψ 2 ξ = χ \{0} (ξ) , ∀ ξ ∈ R its three nontrivial frequency [f] ˙ s,p  2 k∆jfk p , continuous X → Y . K (t, x) is then the y -intercept of of the structural theorem relating these two Z R Bq L j∈ q 0 localizations ∆ f, j ∈ {0, ±4}. Z ` (Z) Readily observed is that K (·, x) is tangent to ∂Γ(x) with slope −1/t [BL76]. sequence of functions with pairwise disjoint −j  ˆ∨ P j A central component of interpolation interpolation methods. ∆jf = ψ 2 · f , f ∆jf. where ∼ j∈Z q/p !1/q theory is the search for interpolation concave and increasing for each x. The sum characterization holds: Fourier transforms, then they are Z Z |f(x + h) − f(x)|p  dh define the s, q-real : The ‘ ’ is to signify that both sides of localizations behave orthogonally [f] ˙ s,p = sp dx n , The real method of interpolation (σ) orthogonal in the sense that ∼ Bq methods. These are general ‘constructions’ n o n n |h| |h| (X0,X1)s,q = X1+(X0,X1)s,q . 2 the expression essentially encode the in various functional settings. In R R generates interpolation spaces by (X0,X1)s,q = x ∈ X0 + X1 | [x]s,q < ∞ P P 2 which when given an interpolation pair with equivalent seminorms, where the fj = kfjk 2 . (3) [Gra14a,Gra14b]. j 2 j L same information up to the addition of a doing so, one obtains a plethora of A = {A ,A } produce a seminormed imposing conditions on this function’s Z 1/q L 0 1  −s q dt right space is given the usual If we replace 2 with p, then in general we polynomial. frequency characterizations. We space F (A) in such a way that if asymptotics. Specifically, for [x]s,q = t K (t, x) . + t seminorm on the sum [ST20]. cannot even say that both sides of the The next step in the Littlewood-Paley now list the ones relevant to this B = {B ,B } is another pair then 0 < s < 1 and 1 ≤ q ≤ ∞ we R 0 1 equation are comparable. theory is to examine how these research.

i.e. they admit well-defined distributional standard techniques in harmonic analysis. in the screened space seminorm is quite nice: Hence we can substitute K for ωp in Our paper containing the full details Z Z p 1/p Fourier transforms. Also, the sum  |f (x + h) − f (x)| dh  expression (4) and deduce the truncated of these results and more can be found THEOREM #1: [f]W˜ s,p = sp dx n . THEOREM #2: COROLLARY 2.1: (σ) n |h| |h| decomposition gives us a hint at how we can (0,σ) R interpolation characterization. at https://arxiv.org/abs/2003.12518. stitch together the Fourier characterizations of high/low decompostion Using standard Hardy-like inequalities, we Interpolation COROLLARY 1.1: Frequency Space the two factors. Formally, the seminorm on Let ϕ be a smooth, compactly supported, show that the above expression is equivalent sum decompostion the sum generates the strongest possible and radial funtion such that ϕ = 1 on the Characterization to one that is directly measuring asymptotics Characterization p By applying our abstract sum vector topology subject to the constraint of unit ball. Then we have the following of the L modulus of continuity, ωp: continuous inclusion mappings of both factors Z 1/p characterization of general truncated equivalent seminorm on the screened n  −s p dt n Let σ : → (0, ∞] be a screening [f]W˜ s,p  t ωp (t, f) . (4) Let σ : R → (0, ∞] be a screening into the sum. Sobolev space (with parameters as in the R (σ) t interpolation spaces, we learn that the (0,σ) function, bounded above and below by If we look to the Littlewood-Paley frequency theorem statment): function, bounded above and below by This latter modulus quantifies the strong screened Sobolev spaces can be realized as positive contstants. If 0 < s < 1 and p the sum of a pair of well-known function positive contstants. If 0 < s < 1 and space characterizations of the homogeneous ∨ continuity in L of the translation group 1,p  ˆ 1 ≤ p < ∞, then the screened Sobolev 1 < p < ∞, then the screened Sobolev Sobolev space W˙ and the [f] ˜ s,p  ∇ ϕf acting on the space of locally integrable spaces: W(σ) ˜ s,p n space ˜ s,p n admits the following ˙ s,p W (R ) is topologically functions, e.g.: s,p n 1,p n s,p n W(σ) (R ) homogeneous Besov Space Bp we (σ) ˜ ˙ ˙ ∨ W(σ) (R ) = W (R )+Bp (R ) .   indistinguishable from a truncated ωp (t, f) = sup {kf (· + h) − fkLp : 0 ≤ |h| ≤ t} . Fourier space characterization observe that the former space has less ˆ ˆ 1/2 + f − ϕf . p n  2  control than the latter over the lower s,p interpolation space between L (R ) Next, we look for inspiration from the theory P j  Bp COROLLARY 1.2: [f]W˜ s,p  j∈ \ 2 |∆jf| and W˙ 1,p ( n), e.g: of inhomogeneous Besov spaces. This scale (σ) Z N Lp frequencies |ξ| ≤ 1. On the other hand, the R (1) dense subspaces s,p n  p n 1,p n  1/p opposite is true over the high frequencies. Multiplication by ϕ and 1 − ϕ on the Fourier W˜ ( ) = L ( ) , W˙ ( ) of spaces is generated via Peetre’s real  P sj p  (σ) R R R + 2 k∆jfk p , side are high and low pass filters, respectively. s,p method of interpolation. They’re ‘inbetween’ As an immediate consequence of the sum j∈N L Hence a natural guess for a frequency space as vector spaces and their seminorms Hence this result is saying precisely that that the end points Lp and W 1,p. Since the decomposition, we know exactly when ˜ s,p n characterization of the seminorm on their sum generate the same topology. for all f ∈ W(σ) (R ). I.e. the topology ˙ 1,p the low mode behavior - the growth at infinity - ˙ 1,p smooth functions and compactly is to in W on the low modes screened spaces are connected to W generalted by the expression on the ˙ s,p of members of these screened Sobolev We proved this result by first studying the supported smooth functions are dense. and Bp on the high modes. via trace theory, we decided to study the K right hand side is no different from the spaces is no worse than general functions case when the screening function is a positive The latter is dense for all 1 ≤ p < ∞ , In rigorously proving this Fourier functional on the sum Lp + W˙ 1,p. original. with gradient in Lp. On the other hand the constant and then used this to deduce the 0 < s < 1, and n ≥ 1; on the other hand, characterization, the sum decomposition and Remarkably, we found that this K functional high mode behavior - the local smoothness - result for the general screening as in the the former is a dense subspace if and only if We first note that because of the sum the intuition about the seminorm on the sum is an equivalent measurement of the is no worse than general functions belonging statement. In the former scenario, the 1 < p or n ≥ 2. decomposition one obtains that members of do all the hard work for us. The estimates s,p Lp-modulus of continuity. to the inhomogeneous Besov space, B . expression obtained by swapping integrals these screened Sobolev spaces are tempered which make the intuition precise follow from p