Generalized Besov-type and Triebel–Lizorkin-type spaces
Volume 273 / 2023
Studia Mathematica 273 (2023), 161-199
MSC: Primary 46E35; Secondary 42B35.
DOI: 10.4064/sm230218-4-9
Published online: 27 November 2023
Abstract
Let $0 \lt p \lt \infty $, $0 \lt 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 $t \gt 0$. We establish several properties of these spaces, including some embedding properties. We also obtain the atomic decomposition of the spaces $B_{p,q}^{s,\varphi }(\mathbb {R}^d)$ and $F_{p,q}^{s,\varphi }(\mathbb {R}^d)$.
