Boundedness of sublinear operators in Triebel–Lizorkin spaces via atoms

Liguang Liu ¹   ; Dachun Yang ¹

¹ School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 People's Republic of China

Studia Mathematica, Tome 190 (2009) no. 2, pp. 163-183

Résumé

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.

Détail

DOI : 10.4064/sm190-2-5

Keywords: mathbb infty proved sublinear operator nbsp uniquely extends bounded sublinear operator triebel lizorkin space dot mathbb quasi banach space mathcal only sup mathcal mbox infinitely differentiable mbox atom dot mathbb infty where atom dot mathbb defined han paluszy ski weiss