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Sharp \(L^2\) estimates of the Schrödinger maximal function in higher dimensions. (English) Zbl 1433.42010

This paper deals mainly with the allmost everywhere convergence of solutions of the free Schrödinger equation.
Consider the following free Schrödinger equation: \[ \begin{cases} i\partial _{t}u-\Delta u=0&\text{on }\mathbb{R}^{n}\times \mathbb{R},\\ u(x,0)=f(x)&\text{on }\mathbb{R}^{n}, \end{cases}\tag{1} \] where \(f\) is a given function in some Sobolev space \(H^{s}(\mathbb{R}^{n})\).
An old question of L. Carleson [Lect. Notes Math. 779, 5–45 (1980; Zbl 0425.60091)] asks whether the solution of (1), namely \(u(x,t)=e^{it\Delta}f(x)\) converges almost everywhere to \(f(x)\) when \(t\rightarrow 0\). A first negative result in this direction was proved by J. Bourgain in [J. Anal. Math. 130, 393–396 (2016; Zbl 1361.35151)], showing that if \(s<\) \(\frac{n}{2(n+1)}\), then the convergence can fail. In the considered paper it is proved the sharpness of the exponent \(\frac{n}{2(n+1)}\), in the sense that, if \(s>\frac{n}{2(n+1)}\) then \(e^{it\Delta}f(x)\) converges almost everywhere to \(f(x)\) on \(\mathbb{R}^{n}\). The result is of great importance, since it settles almost completely the question of Carleson (the case \(s=\frac{n}{2(n+1)}\) does not seem to be covered anywhere).
In the present paper, this convergence result is proved to be a consequence of a “fractal \(L^{2}\) restriction estimate”, which is the the main result of the paper. In order to state this, we fix some notation and terminology.
Any cube of the form \(l+[0,L]^{n}\) where \(L>0\) and \(l\in (L\mathbb{Z})^{n}\) will be called a lattice \(L\)-cube. By \(B^{d}(0,1)\) we denote the unit ball in \(\mathbb{R}^{d}\) (both values \(d=n\) for \(\mathbb{R}^{n}\), and \(d=n+1\) for \(\mathbb{R}^{n}\times \mathbb{R}\), will be used). Suppose now that \(R\geq 1\), \(\lambda \geq 1\) are given and \[ X_{R,\lambda}=\bigcup_{k\in \mathcal{F}}B_{k} \] is a union of lattice \(1\)-cubes \(B_{k}\) such that \(X_{R,\lambda}\subset B^{n+1}(0,R)\) and each lattice \(R^{1/2}\)-cube contains \(\lambda\) many cubes \(B_{k}\) from \(X_{R,\lambda}\). Fix a real number \(1\leq \alpha\leq n+1\). To the set \(X_{R,\lambda}\) we associate the following quantity \[ \gamma _{\alpha}:=\max_{\substack{ B^{n+1}(x^{\prime},r)\subset B^{n+1}(0,R) \\ x^{\prime}\in \mathbb{R}^{n}\times \mathbb{R},r\geq 1}} \frac{\vert \{ k\in \mathcal{F}\mid B_{k}\subset B^{n+1}(x^{\prime},r)\}\vert}{r^{\alpha}}. \]
With this we have:
Theorem 1 (adapted after Theorem 1.6): Suppose \(\varepsilon >0\) is given. Let \(f\in L^{2}(\mathbb{R}^{n})\) be a function with \(\operatorname{supp}\widehat{f} \subset B^{n}(0,1)\). If \(R\geq 1\), \(\lambda \geq 1\), \(1\leq \alpha \leq n+1\) and \(X_{R,\lambda}\) is as above, then \[ \left\Vert e^{it\Delta}f\right\Vert _{L^{2}(X_{R,\lambda})}\leq C_{\varepsilon}\gamma^{\frac{2}{(n+1)(n+2)}}\lambda^{\frac{n}{(n+1)(n+2)}}R^{\frac{\alpha}{(n+1)( n+2)}+\varepsilon}\left\Vert f\right\Vert_{L^{2}(\mathbb{R}^{n})}, \] where \(C_{\varepsilon}\) is a constant only depending on \(\varepsilon\).
There are also discussed some other very interesting consequences of this estimate. These are related to “Hausdorf dimension of the divergence set of Schrödinger solutions”, “Falconer distance set problem”, “Spherical averages Fourier decay rates of fractal measures”. In all these cases it is mentioned that Therem 1 above, implies estimates better than the best known results up this date. We mention below one of these results which is also easy to state.
Suppose \(E\subset \mathbb{R}^{d}\) (\(d\geq 2\)) is a compact set and consider its set of distances \[ \Delta (E) :=\left\{ \left\vert x-y\right\vert \mid x,y\in E\right\} \subset \mathbb{R}. \]
A well-known conjecture of Falconer asserts that, if the Hausdorff dimension of the set \(E\) is strictly greater that \(d/2\), then the Lebesgue measure of \(\Delta(E)\) is positive. In this direction, the considered paper gives a threshold not much greater than \(d/2\): if the Hausdorff dimension of the set \(E\) is strictly greater than \[ \frac{d}{2}+\frac{1}{4}+\frac{1}{8d-4}, \] then the Lebesgue measure of \(\Delta(E)\) is positive.
The most part of the volume of the paper is dedicated to the proof of Theorem 1. The proof is quite technical and uses some tools which are specific to Restriction Theory and Decoupling Theory. Theorem 1 is obtained by induction on scales. In this scope it is proved an inductive (technical) Proposition (Proposition 3.1) from which Theorem 1 is obtained by a particular choice of some parameters and a dyadic pigeonhole principle.
The technique used is similar to those from the works of J. Bourgain and L. Guth [Geom. Funct. Anal. 21, No. 6, 1239–1295 (2011; Zbl 1237.42010)], J. Bourgain and C. Demeter [Ann. Math. (2) 182, No. 1, 351–389 (2015; Zbl 1322.42014)], J. Bourgain [Proc. Steklov Inst. Math. 280, 46–60 (2013; Zbl 1291.35253)], and L. Guth [Acta Math. 221, No. 1, 81–142 (2018; Zbl 1415.42004)]. Also the “multilinear refined Strichartz estimate” obtained in [X. Du et al., Ann. Math. (2) 186, No. 2, 607–640 (2017; Zbl 1378.42011); Forum Math. Sigma 6, Article ID e14, 18 p. (2018; Zbl 1395.42063)] plays an important role.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B37 Harmonic analysis and PDEs
42B25 Maximal functions, Littlewood-Paley theory
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References:

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