Wavelets, Sobolev Multipliers, and Application to Schrödinger Type Operators with Nonsmooth Potentials

Abstract

We employ Meyer wavelets to characterize multiplier space $X_{r,p}^t({ℝ}^n)$ without using capacity. Further, we introduce logarithmic Morrey spaces $M_{r,p}^{t,\tau }({ℝ}^n)$ to establish the inclusion relation between Morrey spaces and multiplier spaces. By fractal skills, we construct a counterexample to show that the scope of the index $\tau$ of $M_{r,p}^{t,\tau }({ℝ}^n)$ is sharp. As an application, we consider a Schrödinger type operator with potentials in $M_{r,p}^{t,\tau }({ℝ}^n)$.