# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## A discrete transform and Triebel-Lizorkin spaces on the bidiscHTML articles powered by AMS MathViewer

by Wei Wang
Trans. Amer. Math. Soc. 347 (1995), 1351-1364 Request permission

## 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.
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