Generalized Besov-type and Triebel-Lizorkin-type spaces

Abstract

Let $0<p<\infty$, $0<q\le \infty$, and $s\in \mathbb{R}$. We introduce a new type of generalized Besov-type spaces $B_{p,q}^{s,\phi }({\mathbb{R}}^d)$ and generalized Triebel-Lizorkin-type spaces $F_{p,q}^{s,\phi }({\mathbb{R}}^d)$, where $\phi$ belongs to the class ${\mathcal{G}}_p$, that is, $\phi :(0,\infty )\to (0,\infty )$ is nondecreasing and $t^{-d/p}\phi (t)$ is nonincreasing in $t>0$. We establish several properties, including some embedding properties, of these spaces. We also obtain the atomic decomposition of the spaces $B_{p,q}^{s,\phi }({\mathbb{R}}^d)$ and $F_{p,q}^{s,\phi }({\mathbb{R}}^d)$.