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

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Characterizations of Screened Sobolev Spaces Noah Stevenson and Ian Tice DEPARTMENTOF MATHEMATICAL SCIENCES,CARNEGIE MELLON UNIVERSITY Characterizations of Screened Sobolev Spaces Noah Stevenson and Ian Tice DEPARTMENT OF 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 space 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 2 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 ^ 2 2 central to the modeling of natural space associated to a Sobolev space on a domain is a space of functions holding Can the screened Sobolev spaces be [f]W~ s;2 minfjξj ; jξj g 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 derivative. Interpolation theory is a set of tools which This expression suggests that the high-mode part of a prominent place in mathematics 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 jrfj < 1 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 seminorm: 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 function 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 jf (x) − f (y)j 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 jx − yj 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; 1] 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 = fx = (x ; xn+1) 2 R j η−(x ) < xn+1 < η+(x )g ⊂ 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 < 1, 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 jrfjp < 1, 1 ≤ p < 1. 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 Fourier transform - 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 jf (x) − f (y)j 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 2 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)) jx − yj 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 < 1 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 j2 j∆jfj : 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 2 f0; 1g are called a relevant to the analysis of screened variant. For σ 2 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. generalized function, as an infinite ∆−4f k;p n a vector space X such that X ⊆ X . W_ for 1 < p < 1 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 2 Given another interpolation couple Yi for (X ;X ) = x 2 X + X j [x] < 1 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 2 f0; 1g 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 r f 2 j∆jfj : y0 dt Lp j2 seminormed spaces X ⊆ X and Let (X ; [·] ) for i 2 f0; 1g be (K (t; x) ; 0) (σ) −s q ∆ f Z Lp i i [x]s;q = t K (t; x) : smoothness 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 Besov space 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 < 1, geometric interpretation of the K function. jξj ≈ 2 . More explicitly, there is a q R and the restriction to Xi is continuous 1-parameter family of seminorms: about the small t behavior, a special remarkable. The above depicts the 1 ≤ q ≤ 1, and 0 < s < 1 has For some x 2 X + X we set special smooth function supported in X ! Y for i 2 f0; 1g 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 ≤ jξj ≤ 2 with of a band limited function f into that TX ⊆ Y and the restriction to X is Γ(x) = y 2 R j 9x =.
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