Keith M. Rogers

Spanish National Research Council | CSIC · Institute of Mathematical Sciences

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of t...

For potentials $V\in L^\infty(\mathbb{R}^2,\mathbb{R})$ and $A\in W^{1,\infty}(\mathbb{R}^2,\mathbb{R}^2)$ with compact support, we consider the Schr\"odinger equation $-(\nabla +iA)^2 u+Vu=k^2u$ with fixed positive energy $k^2$. Under a mild additional regularity hypothesis, and with fixed magnetic potential $A$, we show that the scattering soluti...

We consider a higher dimensional version of the Benjamin--Ono equation, $\partial_t u -\mathcal{R}_1\Delta u+u\partial_{x_1} u=0$, where $\mathcal{R}_1$ denotes the Riesz transform with respect to the first coordinate. We first establish space--time estimates for the associated linear equation, many of which are sharp. These estimates enable us to...

We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's strengthened formulation of the maximal conjecture, concerning families of tubes that satisfy the polynomial Wolff axioms. Our results give improved estim...

We confirm a conjecture of Guth concerning the maximal number of $\delta$-tubes, with $\delta$-separated directions, contained in the $\delta$-neighborhood of an algebraic variety. Modulo a factor of $\delta^{-\varepsilon}$, we also prove Guth and Zahl's generalized version for semialgebraic sets. Although the applications are to be found in harmon...

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Ka...

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 ha...

We consider Carleson’s problem regarding convergence for the Schrödinger equation in dimensions d≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d\ge 2}$$\end{docume...

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 m...

We reconstruct compactly supported potentials with only half a derivative in L-2 from the scattering amplitude at a fixed energy. For this we draw a connection between the recently introduced method of Bukhgeim, which uniquely determined the potential from the Dirichlet-to-Neumann map, and a question of Carleson regarding the convergence to initial...

We introduce a notion of lacunarity in higher dimensions for which we can bound the associated directional maximal operators in LP(ℝn), with p> 1. In particular, we are able to treat the classes previously considered by Nagel-Stein-Wainger, Sjögren-Sjölin and Carbery. Closely related to this, we find a characterization of the sets of directions whi...

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 $s\ge \frac{n}{2(n+2)}$.

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 scattering transform was established by Beals and Coifman for Schwartz functions. Sung extended the validity of the identity to functions belon...

We consider spherical averages of the Fourier transform of fractal measures and improve the lower bound 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 consequenc...

We prove uniqueness for the Calderón 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}...

We prove a sharp bilinear estimate for the one-dimensional Klein–Gordon equation. The proof involves an unlikely combination of five trigonometric identities. We also prove new estimates for the restriction of the Fourier transform to the hyperbola, where the pullback measure is not assumed to be compactly supported.

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on \(L^p(\mathbb {R}^3)\) with \(1 .

Given a discrete group $\G$ and an orthogonal action $\gamma: \G \to O(n)$ we study $L_p$ convergence of Fourier integrals which are frequency supported on the semidirect product $\R^n \rtimes_\gamma \G$. Given a unit $u \in \R^n$ and $1 < p \neq 2 < \infty$, our main result shows that the twisted (directional) Hilbert transform $H_u \rtimes_\gamma...

This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and associated maximal functions. It focuses on the Littlewood–Paley bounds for two square functions introduced by Stein, who had stressed their importance in harm...

We prove sharp Morawetz estimates – global in time with a singular weight in the spatial variables – for the linear wave, Klein-Gordon, and Schrödinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.

We prove a localization principle for directional maximal operators in Lp(n), with p > 1. The resulting bounds, which we conjecture hold for the largest possible class of directions, imply Lebesgue-type differentiation of integrals over tubes that point in the given directions.

We begin with an overview on square functions for spherical and Bochner-Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner-Riesz operator, and for more general classes of radia...

We consider Calderon's inverse problem on planar domains with conductivities in fractional Sobolev spaces. When is Lipschitz, the problem was shown to be stable in the L-2-sense in Clop et al. [Stability of calderon's inverse conductivity problem in the plane for discontinuous conductivities, Inverse Probl. Imaging 4 (2010), 49-91]. We remove the L...

We consider the Schrödinger equation associated to the harmonic oscillator, it∂u=Hui∂tu=Hu, where H=−Δ+2|x|H=−Δ+|x|2, with initial data in the Sobolev space Hs(Rd)Hs(Rd). With d=2d=2 and s>3/8s>3/8, we prove almost everywhere convergence of the solution to its initial data as time tends to zero, which improves a result of Yajima (1990) [30]. To thi...

We consider the Schrödinger equation i∂ t u+Δu=0 with initial data in H s (ℝ n ). A classical problem is to identify the exponents s for which u(·,t) converges almost everywhere to the initial data as t tends to zero. In one spatial dimension, Carleson proved that the convergence is guaranteed when s=1 4, and Dahlberg and Kenig proved that divergen...

We consider the Schrödinger equation for the harmonic oscillator i ∂ t u=Hu, where H=−Δ+|x|2, with initial data in the Hermite–Sobolev space H −s/2L 2(ℝn ). We obtain smoothing and maximal estimates and apply these to perturbations of the equation and almost everywhere convergence problems.

We prove mixed norm space-time estimates for solutions of the Schroedinger equation, with initial data in $L^p$ Sobolev or Besov spaces, and clarify the relation with adjoint restriction.

We refine results of Carleson, Sjögren and Sjölin regarding the pointwise convergence to the initial data of solutions to the Schrödinger equation. We bound the Hausdorff dimension of the sets on which convergence fails. For example, with initial data in H1(\mathbbR3){H^1(\mathbb{R}^{3})}, the sets of divergence have dimension at most one.

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in Lxq(Rn,Ltr(R)) with initial data f in the homogeneous Sobolev space H˙s(Rn). The exponents and regularity index satisfy n+1q+1r=n2 and s=n2−nq−2r. For n=2 we prove the estimates in the range q>16/5, and for n⩾3 in the range q>2+4/(n+1).

We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the $L^4_{t,x}(\R^{5+1})$ norm of the solution in terms of the energy. We also characterise the maximisers.

We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schr\"odinger equations.

We prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators and generalized Radon transforms.

We consider the Schrödinger operator eitD{e^{it\Delta}} acting on initial data f in [(H)\dot]s{\dot{H}^s}. We show that an affirmative answer to a question of Carleson, concerning the sharp range of s for which limt® 0eitDf(x)=f(x){\lim_{t\to 0}e^{it\Delta}f(x)=f(x)} a.e. x Î \mathbb Rn{x\in \mathbb {R}^n}, would imply an affirmative answer to a...

We show that the Schrödinger operator ei t Δ is bounded from Wα, q (Rn) to Lq (Rn × [0, 1]) for all α > 2 n (1 / 2 - 1 / q) - 2 / q and q ≥ 2 + 4 / (n + 1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0 < t < 1 | ei t Δ f | is bounded from Hs (Rn) to Lloc2 (Rn) when s > s0 if and onl...

For alpha > 1 we consider the initial value problem for the dispersive equation i partial derivative(t)u + (-Delta)(alpha/2)u = 0. We prove an endpoint L(p) inequality for the maximal function sup(t is an element of[0, 1]) |u(., t)| with initial values in L(p)-Sobolev spaces, for p is an element of (2 + 4/(d + 1), infinity). This strengthens the fi...

The wave equation, ∂tt u=Δu, in ℝn+1, considered with initial data u(x,0)=f∈H s (ℝn ) and u’(x,0)=0, has a solution which we denote by \(\frac{1}{2}(e^{it\sqrt{-\Delta}}f+e^{-it\sqrt{-\Delta}}f)\). We give almost sharp conditions under which \(\sup_{0<t<1}|e^{\pm it\sqrt{-\Delta}}f|\) and \(\sup_{t\in\mathbb{R}}|e^{\pm it\sqrt{-\Delta}}f|\) are bou...

Let n⩾3 and consider the subcritical nonlinear Schrödinger equation, i∂tu+Δu=|u|αu, with initial data u0∈Hs(Rn). When s⩾1, Kato proved that if a maximal solution exists, then it is unique in C([0,Tmax),Hs). Previously, uniqueness had only been proven in strictly smaller subspaces. The existence of a solution is assured when s∈[0,1], so that the sub...

The Schrodinger equation, i theta(t)u + Delta u = 0, with initial datum f contained in a Sobolev space H-s(R-n), has solution e(it Delta) f. We give sharp conditions under which supt vertical bar e(it Delta) f vertical bar is bounded from H-s(R) to L-q (R) for all q, and give sharp conditions under which sup(0 <= t < 1) vertical bar e(it Delta) f v...

We consider the nonlinear nonelliptic Schrödinger equation defined by i∂tu+(∂x2−∂y2)u+γ|u|2u=0 with initial datum in L2(R2). We show that if the solution blows up in finite time, then there is a mass concentration phenomenon near the blow-up time. The key ingredient is a refinement of the Strichartz inequality on the saddle.

It is conjectured that the solution to the Schrödinger equation in ℝn+1 converges almost everywhere to its initial datum f, for all f ∈ Hs (ℝn), if and only if s ≥ 1/4. It is known that there is an s < 1/2 for which the solution converges for all f ∈ Hs(ℝ2). We show that the solution to the nonelliptic Schrödinger equation, i∂tu + (∂x2 - ∂y2)u = 0,...

A K^n_2-set is a set of zero Lebesgue measure containing a translate of every plane in an (n-2)-dimensional manifold in Gr(n,2), where the manifold fulfills a curvature condition. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dim_H(E)\ge 7/2 for all K^4_2-sets E. When the underlying field is replaced...

Let p denote the p-adic numbers. We consider curves in defined by p-adic polynomials of one p-adic variable. We show that maximal averages along these curves are bounded, where 1 < q < ∞.

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...

We prove a version of van der Corput's Lemma for polynomials over the p-adic numbers.