Weighted estimates for powers and smoothing estimates of Schrödinger operators with inverse-square potentials

Let \La be a Schr\"odinger operator with inverse square potential $a|x|^{-2}$ on \Rd, d\geq 3. The main aim of this paper is to prove weighted estimates for fractional powers of \La. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to \La. As an application, we obtain smoothing estimates regarding the propagator e^{it\La}

Weighted estimates for powers and smoothing estimates of Schrödinger operators with inverse-square potentials

Let \La be a Schr\"odinger operator with inverse square potential $a|x|^{-2}$ on \Rd, d\geq 3. The main aim of this paper is to prove weighted estimates for fractional powers of \La. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to \La. As an application, we obtain smoothing estimates regarding the propagator e^{it\La}

Kato smoothing and Strichartz estimates for wave equations with magnetic potentials

Let H be a selfadjoint operator and A a closed operator on a Hilbert space H. If A is H-(super)smooth in the sense of Kato-Yajima, we prove that AH^(-1/4) is H^(1/2)-(super)smooth. This allows us to include wave and Klein-Gordon equations in the abstract theory at the same level of generality as Schrödinger equations. We give a few applications and in particular, based on the resolvent estimates of Erdogan, Goldberg and Schlag (Forum Mathematicum 21:687–722, 2009), we prove Strichartz estimates for wave equations perturbed with large magnetic potentials on Rn

Low regularity solutions for the wave map equation into the 2-D sphere

A class of weak wave map solutions with initial data in Sobolev space of order s<1 is studied. A non uniqueness result is proved for the case, when the taro,et manifold is a two dimensional sphere. Using an equivariant wave map ansatz a family of self-similar solutions is constructed. This construction enables one to show ill-posedness of the inhomogeneous Cauchy problem for wave maps

Strichartz and smoothing estimates for dispersive equations with magnetic potentials

We prove global smoothing and Strichartz estimates for the Schrodinger, wave, Klein-Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical

Weighted Lp estimates for powers of selfadjoint operators

We prove Lp and weighted Lp estimates for bounded functions of a selfadjoint operator satisfying a pointwise Gaussian estimate for its heat kernel. As an application, we obtain weighted estimates for fractional powers of an electromagnetic Schrödinger operator with singular coefficients. The proofs are based on a modification of techniques due to Hebisch (1990) [17] and Auscher and Martell (2006) [4]. © 2011 Elsevier Inc.

Evolution Equations on Non-Flat Waveguides

We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator H=-Delta(x)-Delta(y)+V(x,y) with Dirichlet boundary conditions on an unbounded domain Omega, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables, x. If Omega is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu-lambda u = f. As consequences, we prove smoothing estimates for the Schrodinger and wave equations associated to H, and Strichartz estimates for the Schrodinger equation. Additionally, we deduce that the operator H does not admit eigenvalues

On the continuity of the solution operator to the wave map system

We investigate the continuity properties of the solution operator to the wave map system from R × Rn to a general nonflat target of arbitrary dimension, and we prove by an explicit class of counterexamples that this map is not uniformly continuous in the critical norms on any neighborhood of 0

Endpoint estimates and global existence for the nonlinear Dirac equation with potential

We prove endpoint estimates with angular regularity for the wave and Dirac equations perturbed with a small potential. The estimates are applied to prove global existence for the cubic Dirac equation perturbed with a small potential, for small initial H-1 data with additional angular regularity. This implies in particular global existence in the critical energy space H-1 for small radial data. (c) 2012 Elsevier Inc. All rights reserved