The aim of these notes is to describe some recent results concerning dispersive estimates for principally normal pseu-dodifferential operators. The main motivation for this comes from uniqueâ€¦ (More)

We obtain L p eigenfunction bounds for the harmonic oscillator H = âˆ’âˆ†+x 2 in R n and for other related operators, improving earlier results of Thangavelu and Karadzhov. We also construct suitableâ€¦ (More)

Let L = âˆ’âˆ† âˆ’ W be a SchrÃ¶dinger operator with a potential W âˆˆ L n+1 2 (R n), n â‰¥ 2. We prove that there is no positive eigenvalue. The main tool is an L p âˆ’ L p â€² Carleman type estimate, whichâ€¦ (More)

Two-dimensional deep water waves and some problems in nonlinear optics can be described by various third order dispersive equations, modifying and generalizing the KdV as well as nonlinearâ€¦ (More)

The purpose of this paper is to use semiclassical analysis to unify and generalize L estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend onâ€¦ (More)

The Cauchy problem for the Kadomtsev-Petviashvili-II equation (ut + uxxx + uux)x + uyy = 0 is considered. A small data global well-posedness and scattering result in the scale invariant,â€¦ (More)

In the late 50â€™s and early 60â€™s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merelyâ€¦ (More)

We show the existence of a global unique and analytic solution for the mean curvature flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also showâ€¦ (More)

We present a general theory to study optimal regularity for a large class of nonlinear elliptic systems satisfying general boundary conditions and in the presence of a geometric transmissionâ€¦ (More)