Wavelets, Multiplier spaces and application to Schr\"{o}dinger type operators with non-smooth potentials

TL;DR

This paper uses wavelet techniques to characterize multiplier spaces between Sobolev spaces and introduces logarithmic Morrey spaces, applying these results to Schrödinger operators with non-smooth potentials.

Contribution

It provides a wavelet-based characterization of multiplier spaces and introduces new logarithmic Morrey spaces, establishing their relation and applying to Schrödinger operators.

Findings

Wavelet characterization of multiplier spaces

Introduction of logarithmic Morrey spaces and their properties

Application to Schrödinger operators with non-smooth potentials

Abstract

In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$ to establish the inclusion relation between Morrey spaces and multiplier spaces. By wavelet characterization and fractal skills, we construct a counterexample to show that the scope of the index $\tau$ of $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$ is sharp. As an application, we consider a Schr\"odinger type operator with potentials in $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$.