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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A discrete transform and Triebel-Lizorkin spaces on the bidisc


Author: Wei Wang
Journal: Trans. Amer. Math. Soc. 347 (1995), 1351-1364
MSC: Primary 46E35; Secondary 42B20, 46F05
MathSciNet review: 1254857
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1254857-8
PII: S 0002-9947(1995)1254857-8
Article copyright: © Copyright 1995 American Mathematical Society