Fourier Multipliers between Weighted Anisotropic Function Spaces. Part I: Besov Spaces

Abstract

We determine classes $M(B_{p_0,q_0}^{s_0}(w_0),B_{p_1,q_1}^{s_1}(w_1))$ of Fourier multipliers between weighted anisotropic Besov spaces $B_{p_0,q_0}^{s_0}(w_0)$ and $B_{p_1,q_1}^{s_1}(w_1)$ where $p_0\le 1$ and $w_0,w_1$ are weight functions of, polynomial growth. To this end we use a discrete characterization of the function spaces akin to the $\phi$-transform of Frazier and Jawerth which leads to a unified approach to the multiplier problem. In this way widely generalized versions of known results of Bui, Johnson, Peetre and others are obtained from a single theorem.