A discrete transform and Triebel-Lizorkin spaces on the bidisc

Abstract:

We use a discrete transform to study the Triebel-Lizorkin spaces on bidisc $\dot F_p^{\alpha q}, \dot f_p^{\alpha q}$ and establishes the boundedness of transform ${S_\phi }:\dot F_p^{\alpha q} \to \dot f_p^{\alpha q}$ and ${T_\psi }:\dot f_p^{\alpha q} \to \dot F_p^{\alpha q}$. We also define the almost diagonal operator and prove its boundedness. With the use of discrete transform and Journé lemma, we get the atomic decomposition of $\dot f_p^{\alpha q}$ for $0 < p \leqslant 1, p \leqslant q < \infty$. The atom supports on an open set, not a rectangle. Duality ${(\dot f_1^{\alpha q})^{\ast }} = \dot f_\infty ^{ - \alpha q’}, \tfrac {1} {q} + \tfrac {1} {{q’}} = 1, q > 1, \alpha \in R$, is established, too. The case for $\dot F_p^{\alpha q}$ is similar.