Keith M. Rogers

GLOBAL UNIQUENESS FOR THE CALDERÓN PROBLEM WITH LIPSCHITZ CONDUCTIVITIES

We prove uniqueness for the Calderon 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

A sharp Strichartz estimate for the wave equation with data in the energy space

C^{1}

Improved bounds for Stein's square functions

-conductivities and Lipschitz conductivities sufficiently close to the identity.

Endpoint maximal and smoothing estimates for Schrödinger equations

We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the

A Calderón–Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators

L^4_{t,x}(\R^{5+1})

On the size of divergence sets for the Schroedinger equation with radial data

norm of the solution in terms of the energy. We also characterise the maximisers.

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

Supported in part by NRF grant 2009-0072531 (Korea), MICINN grant MTM2010-16518 (Spain), ERC grant 277778 (Europe), and NSF grant 0652890 (USA).

On the polynomial Wolff axioms

Abstract For α > 1 we consider the initial value problem for the dispersive equation i∂tu + (–Δ) α/2 u = 0. We prove an endpoint Lp inequality for the maximal function with initial values in Lp -Sobolev spaces, for p ∈ (2 + 4/(d + 1), ∞). This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp Lp space-time estimates (local in time) for the same range of p.

A note on pointwise convergence for the Schrödinger equation

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [10] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator with those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques. In the present article we fill the gap left open in [10]. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed in [9] for a particular critical operator on the binary tree. Those problems appeared in natural way during the study of compactness properties of certain Volterra integral operators in a critical case.

Chapter Twelve. Square Functions and Maximal Operators Associated with Radial Fourier Multipliers

This research has been supported by EPSRC grant EP/E022340/1, ERC grant 277778, and MINECO grants SEV-2011-0087 and MTM2010-16518.