Document Type

Conference Proceeding

Publication Date

2008

Publication Title

Contemporary Mathematics in AMS Book Series

DOI

10.1090/conm/464

ISBN

978-0-8218-8143-9

Abstract

In this article, we give an overview of some recent developments in Littlewood-Paley theory for Schrödinger operators. We extend the Littlewood-Paley theory for special potentials considered in our previous work [J. Fourier Anal. Appl. 12 (2006), no. 6, 653–674; MR2275390]. We elaborate our approach by considering a potential in C0 or the Schwartz class in one dimension. In particular, the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give a maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with H. Then we prove a spectral multiplier theorem on these spaces and derive Strichartz estimates for the wave equation with a potential. We also consider a similar problem for the unbounded potentials in the Hermite and Laguerre cases, whose potentials V=a|x|2+b|x|−2 are known to be critical in the study of perturbation of nonlinear dispersive equations. This improves upon the previous results when we apply the upper Gaussian bound for the heat kernel and its gradient.

Comments

This version of the paper was obtained from arXIV.org. In order for the work to be deposited in arXIV.org, the author must have permission to distribute the work or the work must be available under the Creative Commons Attribution license, Creative Commons Attribution-Noncommercial-ShareAlike license, or Create Commons Public Domain Declaration. The publisher's final edited version of this article is available at Contemporary Mathematics.

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