H\"older Inequalities for Besov-Morrey and Triebel-Lizorkin-Morrey Spaces

Abstract

In this paper we prove H\"older inequalities for the Besov-Morrey spaces $ \mathcal{N}^{s}_{u,p,q}(\mathbb{R}^{d}) $ and the Triebel-Lizorkin-Morrey spaces $ \mathcal{E}^{s}_{u,p,q}(\mathbb{R}^{d}) $. We observe that these Morrey smoothness spaces are pointwise multiplier spaces if certain conditions concerning the parameters also including $ s>d \max ( 0, \frac{1}{p} - 1) $ are fulfilled. In this context we significantly generalize the H\"older inequalities for the original Besov and Triebel-Lizorkin spaces obtained by Sickel and Triebel in 1995. For the proofs we use paramultiplication and some Franke-Jawerth embeddings obtained by Haroske and Skrzypczak.