Jonathan Bennett

Stability of the Brascamp-Lieb constant and applications

abstract:We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis.

Optimal control of singular Fourier multipliers by maximal operators

We control a broad class of singular (or “rough”) Fourier multipliers by geometrically-defined maximal operators via general weighted L(R) norm inequalities. The multipliers involved are related to those of Coifman– Rubio de Francia–Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form ei|ξ| α for both α positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more novel “improper fractional averages” associated with “escape” regions. Some applications are given to the theory of Lp − Lq multipliers, oscillatory integrals and dispersive PDE, along with natural extensions to higher dimensions. Dedicated to the memory of Adela Moyua, 1956–2013.

Strichartz inequalities with weights in Morrey-Campanato classes

We prove some weighted refinements of the classical Strichartz inequalities for initial data in the Sobolev spaces Ḣs(ℝn). We control the weightedL2-norm of the solution of the free Schrödinger equation whenever the weight is in a Morrey-Campanato type space adapted to that equation. Our partial positive results are complemented by some necessary conditions based on estimates for certain particular solutions of the free Schrödinger equation.

The Fourier extension operator on large spheres and related oscillatory integrals

We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds showing sharpness are proved using Kakeya set constructions. As a special case of the upper bounds we deduce optimal

Heat-flow monotonicity related to the Hausdorff-Young inequality

L^p(mathbb{S}^2)\to L^q(R \mathbb{S}^2)

Subdyadic square functions and applications to weighted harmonic analysis

estimates for the Fourier extension operator on large spheres in

On the Strichartz Estimates for the Kinetic Transport Equation

\mathbb{R}^3

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

, which are uniform in the radius

Behaviour of the Brascamp–Lieb constant

R

Transversal multilinear Radon-like transforms: local and global estimates.

. Two appendices are included, one concerning an application to Lorentz space bounds for averaging operators along curves in