Harmonic analysis related to Schroedinger operators

TL;DR

This paper reviews recent advances in Littlewood-Paley theory for Schr"odinger operators, including new asymptotics, function space characterizations, spectral multipliers, and Strichartz estimates, extending previous work to more general potentials.

Contribution

It extends Littlewood-Paley theory to broader classes of potentials, providing refined eigenfunction asymptotics, new function space characterizations, and improved estimates for Schr"odinger operators.

Findings

Refined asymptotics for eigenfunctions and Fourier transforms.

Maximal function characterization of associated Besov and Triebel-Lizorkin spaces.

Spectral multiplier theorem and Strichartz estimates for wave equations with potentials.

Abstract

In this article we give an overview on some recent development of Littlewood-Paley theory for Schr\"odinger operators. We extend the Littlewood-Paley theory for special potentials considered in the authors' previous work. We elaborate our approach by considering potential in $C^\infty_0$ or 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 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 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.