Hölder Inequalities and Sharp Embeddings in Function Spaces of $B_{pq}^s$ and $F_{pq}^s$ Type

Abstract

Besides assertions on sharp embeddings of $B_{pq}^s$ and $F_{pq}^s$ we give necessary and sufficient conditions on the parameters $s,p,q,p_1,q_1,p_2,q_2$ for the Hölder type inequalities $\Vert f\cdot g|B_{pq}^s\Vert \le c\Vert f|B_{p_1{q}_1}^s\Vert \Vert g|B_{p_2{q}_2}^s$ and $\Vert f\cdot g|F_{pq}^s\Vert \le c\Vert f|F_{p_1{q}_1}^s\Vert \Vert g|F_{p_2{q}_2}^s$ to hold.