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

In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M_{r,p}^{t,\tau }({\mathbb{R}}^n)$ 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 $M_{r,p}^{t,\tau }({\mathbb{R}}^n)$ is sharp. As an application, we consider a Schr"odinger type operator with potentials in $M_{r,p}^{t,\tau }({\mathbb{R}}^n)$.