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Keith Rogers

Publications and source records attributed to Keith Rogers.

9 recordsLinked to original sources
arXiv2018

Falconer's distance set problem via the wave equation

Falconer proved that there are sets $E\subset \mathbb{R}^n$ of Hausdorff dimension $n/2$ whose distance sets $\{|x-y| : x,y\in E\}$ are null with respect to Lebesgue measure. This led to the conjecture that distance sets have positive Lebesgue measure as soon the Hausdorff dimension of $E$ is larger than $n/2$. The best results in this direction have exploited estimates that restrict the Fourier transform of measures to the $(n-1)$-dimensional sphere. Here we show that these estimates can be replaced by estimates that restrict the Fourier transform of measures to the $n$-dimensional cone. Such estimates were first considered by Wolff in their adjoint form whereby they bound the solution to the wave equation in terms of its initial data. The connection with Falconer's problem, combined with Falconer's counterexample, provides a new necessary condition for what was considered a plausible conjecture for these estimates.

math.CA↗
arXiv2017

A note on pointwise convergence for the Schrödinger equation

We consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain recently proved that there is initial data, in $H^s(\mathbb{R}^n)$ with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.

math.CA↗
arXiv2016

Global uniqueness for the Calderón problem with Lipschitz conductivities

We prove uniqueness for Calderón's problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three and four dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^1$ conductivities and Lipschitz conductivities sufficiently close to the identity.

math.AP↗
arXiv2015

Average decay of the Fourier transform of measures with applications

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 Schrödinger 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 $\dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$.

math.CA↗
arXiv2015

An improved necessary condition for the Schrödinger maximal estimate

We improve the necessary condition for Carleson's problem regarding convergence for the Schrödinger equation in dimensions $n\ge 3$. We prove that if the solution converges almost everywhere to its initial datum as time tends to zero, for all data in $H^s(\mathbb{R}^n)$, then $s\ge \frac{n}{2(n+2)}$.

math.CA↗
arXiv2015

On Plancherel's identity for a two-dimensional scattering transform

We consider the $\overline{\partial}$-Dirac system that Ablowitz and Fokas used to transform the defocussing Davey-Stewartson system to a linear evolution equation. The nonlinear Plancherel identity for the associated scattering transform was established by Beals and Coifman for Schwartz functions. Sung extended the validity of the identity to functions belonging to $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ and Brown to $L^2(\mathbb{R}^2)$-functions with sufficiently small norm. More recently, Perry extended to the weighted Sobolev space $H^{1,1}(\mathbb{R}^2)$ and here we extend to $H^{s,s}(\mathbb{R}^2)$ with $s\in(0,1)$.

math.CA↗
arXiv2003

Sharp van der Corput estimates and minimal divided differences

We find the nodes that minimise divided differences and use them to find the sharp constant in a sublevel set estimate. We also find the sharp constant in the first instance of the van der Corput Lemma using a complex mean value theorem for integrals. With these sharp bounds we improve the constant in the general van der Corput Lemma, so that it is asymptotically sharp.

math.CA↗