Hölder Inequalities and Sharp Embeddings in Function Spaces of $B^s_{pq}$ and $F^s_{pq}$ Type

Abstract

Besides assertions on sharp embeddings of $B^s_{pq}$ and $F^s_{pq}$ 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 $\| f \cdot g | B^s_{pq} \| ≤ c \| f | B^s_{p_1q_1} \| \| g | B^s_{p_2q_2}$ and $\| f \cdot g | F^s_{pq} \| ≤ c \| f | F^s_{p_1q_1} \| \| g | F^s_{p_2q_2}$ to hold.