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Wavelets and Geometric Structure for Function Spaces

Authors:	Email author" target="_blank">Qi?Xiang?Yang Email author

Institution:	(1) Department of Mathematics, Wuhan University, Wuhan, 430072, P. R. China

Abstract:	Abstract With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces $\ifmmode\expandafter\dot\else\expandafter\.\fi{B}^{{s,q}}_{p} {\left( {s \in R,0 < p,q \leqslant \infty } \right)}$ and Triebel–Lizorkin spaces $\ifmmode\expandafter\dot\else\expandafter\.\fi{F}^{{s,q}}_{p} {\left( {s \in R,0 < p < \infty ,0 < q \leqslant \infty } \right)}$ ; but the structure of dual spaces $\ifmmode\expandafter\dot\else\expandafter\.\fi{D}^{{s,q}}_{p}$ of $\ifmmode\expandafter\dot\else\expandafter\.\fi{F}^{{s,q}}_{p} {\left( {s \in R,0 < p \leqslant 1 \leqslant q \leqslant \infty } \right)}$ is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize $\ifmmode\expandafter\dot\else\expandafter\.\fi{D}^{{s,q}}_{p} {\left( {s \in R,0 < p = 1 \leqslant q \leqslant \infty } \right)}$ in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among $\ifmmode\expandafter\dot\else\expandafter\.\fi{B}^{{0,q}}_{1} ,{\kern 1pt} {\kern 1pt} \ifmmode\expandafter\dot\else\expandafter\.\fi{F}^{{0,q}}_{1}$ and L ¹ are studied. Supported by NNSF of China (Grant No. 10001027)

Keywords:	Triebel– Lizorkin spaces Dual spaces Wavelets
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