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In this paper, we employ Meyer wavelets to characterize multiplier spacesbetween Sobolev spaces without using capacity. Further, we introducelogarithmic Morrey spaces $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$ to establish theinclusion relation between Morrey spaces and multiplier spaces. By waveletcharacterization and fractal skills, we construct a counterexample to show thatthe 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$M^{t,\tau}_{r,p}(\mathbb{R}^{n})$.

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