Hölder Inequalities for Besov-Morrey and Triebel-Lizorkin-Morrey Spaces

Abstract

In this paper we prove Hölder inequalities for the Besov-Morrey spaces ${\mathcal{N}}_{u,p,q}^s({\mathbb{R}}^d)$ and the Triebel-Lizorkin-Morrey spaces ${\mathcal{E}}_{u,p,q}^s({\mathbb{R}}^d)$. 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 Hölder 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.

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