Average decay of the Fourier transform of measures with applications

By Renato Lucà, Keith McKenzie Rogers. Journal of the European Mathematical Society (Print), 2015

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We consider spherical averages of the Fourier transform of fractal measures and improve both the upper and lower bounds on the rate of decay. Maximal estimates with respect to fractal measures are deduced for the Schr\"odinger and wave equations. This refines the almost everywhere convergence of the solution to its initial datum as time tends to zero. A consequence is that the solution to the wave equation cannot diverge on a $(d-1)$-dimensional manifold if the data belongs to the energy space $\dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/351467b0878f4404ef6eb52decb81e26b394f78c

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Citations in the corpus

Sharp L2 estimates of the Schrödinger maximal function in higher dimensions (2018, result): “measure in Rn+1. Then (2.5) eit f L2(Bn+1(0;R);d R(x;t)) /R 2(n+1) kfk 2 ; whenever R1 and fhas Fourier support in Bn(0;1). Theorem 2.2 and 2.3 are new when n2;2(2;3) or n3;2(n+1 2 ;n+ 1) (c.f. [14, 22]). We defer the proof of these weighted L2 estimates to the end of this subsection. Let’s rst see ”
Weighted restriction estimates and application to Falconer distance set problem (2018, unknown): “ αP p0,1{2s, (Mattila [13]) 1{2, αP r1{2,1s, (Mattila [13]) α{2, αP r1,2s, (Wolﬀ [16]). WEIGHTED RESTRICTION AND FALCONER’S PROBLEM 3 The problem remains open when dě 3 and αą d´1 2 . See Luca-Rogers [12] and the references therein for example for a discussion of various partial results. In higher d”
POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES (2018, background): “ ≥ 3. Then (1.2) αn(s) ≤ n+1−(2 + 2 n+1 )s, n +1 2(n+2) < s < n/4. Remark 1.3. Theorem 1.2 also holds when n = 2 and that recovers the previous results of Lee [13], Bourgain [3] and Luca-Rogers [14], by a diﬀerent method. In [8], the almost sharp result s > 1/3 is obtained in the setting of”
Improved Fourier restriction estimates in higher dimensions (2018, background): “There is certainly a precedent for the recursive approach: for instance, in the fourth section of [6], Bourgain and Guth reformulate their key inductionon-scale argument as a recursive procedure to allow for the use of additional information coming from X-ray transform estimates (see also [22, 37] f”