Characterization of anysotropic Liouville-type mixed spaces and its application

Abstract

Within the framework of the method of hypersingular integrals we obtain a characterization of the anysotropic spaces $L_{\bar p, \bar r}^{\bar \alpha}$. These spaces are defined to consist of functions $f(x)\in L_{\bar r}$ for which $ F^{-1}(\sum_{j=1}^n |\xi_j|^{\alpha_j})Ff\in L_{\bar p}, \alpha_j>0. $ The mentioned characterization is applied to prove the denseness of the class $C_0^\infty$ in $L_{\bar p, \bar r}^{\bar \alpha}$.