Boundedness of sublinear operators in Triebel–Lizorkin spaces via atoms
Volume 190 / 2009
Studia Mathematica 190 (2009), 163-183
MSC: Primary 46E35; Secondary 42B20, 42B25, 47A30.
DOI: 10.4064/sm190-2-5
Abstract
Let $s\in\mathbb R$, $p\in(0, 1]$ and $q\in[p, \infty)$. It is proved that a sublinear operator $T$ uniquely extends to a bounded sublinear operator from the Triebel–Lizorkin space $\dot{F}^s_{p, q}({\mathbb R}^{n})$ to a quasi-Banach space $\mathcal B$ if and only if $$ \sup\{ \|T(a)\|_{\mathcal B}:\, a \mbox{ is an infinitely differentiable } (p, q, s)\mbox{-atom of }\dot{F}_{p,q}^{s}(\mathbb R^n) \}< \infty, $$ where the $(p, q, s)$-atom of $\dot{F}_{p,q}^{s}(\mathbb R^n)$ is as defined by Han, Paluszyński and Weiss.
