Ana Vargas

A bilinear approach to the restriction and Kakeya conjectures

The purpose of this paper is to investigate bilinear variants of the restriction and Kakeya conjectures, to relate them to the standard formulations of these conjectures, and to give applications of this bilinear approach to existing conjectures. The methods used are based on several observations and results of Bourgain (see [2]-[6]), together with some refinements by Moyua, Vargas, and Vega [17, 18]. This paper is organized as follows. In the first section we discuss bilinear restriction estimates, and show how one can pass back and forth between these estimates and the standard restriction estimates. We also generalize the 12/7 bilinear restriction estimate of [18] to higher dimensions. In the second section we give analogues of the above results for the Kakeya operator. In particular we give a bilinear improvement to Wolffs Kakeya theorem in arbitrary dimension. In the third section we give applications of these bilinear estimates in three dimensions. For example, we are able to improve the 42/11 exponent in Wolffs restriction theorem to 34/9. We are also able to prove a sharp (LP, Lq) restriction theorem which improves on the classical (L2, L4) Tomas-Stein theorem, and also give some concrete progress on a bilinear restriction conjecture of Klainerman and Machedon. We also give a non-bilinear approach to these estimates, which gives weaker results but is more direct and probably has a wider range of application.

Mass Concentration Phenomena for the L^2-Critical Nonlinear Schrödinger Equation

In this paper, we show that any solution of the nonlinear Schr{o}dinger equation

Partial Recovery of a Potential from Backscattering Data

iu_t+\Delta u\pm|u|^\frac{4}{N}u=0,

Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite L2 norm

which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on the Bourgains one~\cite{MR99f:35184}, which has established this result in the bidimensional spatial case, and on a generalization of Strichartzs inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega~\cite{MR1671214}. We also generalize to higher dimensions the results in Keraani~\cite{MR2216444} and Merle and Vega~\cite{MR1628235}.

Sharp null form estimates for the wave equation

ABSTRACT We prove that the main singularities,measured in the scale of Sobolev spaces,of the potential q in the Schrödinger Hamiltonian −Δ+q,in dimensions n=2,3,are contained in the Born approximation for backscattering data.

Directional operators and radial functions on the plane

Abstract We prove global wellposedness for the one-dimensional cubic non-linear Schrodinger equation in a space of distributions which is invariant under Galilean transformations and includes L 2 . This space arises naturally in the study of the restriction properties of the Fourier transform to curved surfaces. The L p bounds, p ≠2, for the extension operator, dual to the restricition one, plays a fundamental role in our approach.

Localization for Riesz means of Fourier expansions

Null form estimates (from

A remark on a maximal function over a Cantor set of directions

\dot{H}^{\alpha_1}\times\dot{H}^{\alpha_2}

RECONSTRUCTION OF DISCONTINUITIES FROM BACKSCATTERING DATA IN TWO DIMENSIONS

to

Checkerboards, Lipschitz functions and uniform rectifiability

L^q_t(L^r_x)