Generalized Besov-type and Triebel-Lizorkin-type spaces

Abstract: Let $0<p<\infty$, $0<q\leq\infty$, and $s\in\mathbb{R}$. We introduce a new type of generalized Besov-type spaces $B_{p,q}^{s,\varphi}(\mathbb{R}^d)$ and generalized Triebel-Lizorkin-type spaces $F_{p,q}^{s,\varphi}(\mathbb{R}^d)$, where $\varphi$ belongs to the class $\mathcal{G}_p$, that is, $\varphi:(0,\infty) \rightarrow (0,\infty)$ is nondecreasing and $t^{-d/p}\varphi(t)$ 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 $B_{p,q}^{s,\varphi}(\mathbb{R}^d)$ and $F_{p,q}^{s,\varphi}(\mathbb{R}^d)$.