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Convergence of sequences of Calderón‐Zygmund operators with application to wavelet expansions
Author(s) -
Li Kangwei,
Sun Wenchang
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200159
Subject(s) - mathematics , hardy space , norm (philosophy) , operator (biology) , smoothness , besov space , uniform norm , pure mathematics , operator norm , extrapolation , sequence (biology) , convergence (economics) , mathematical analysis , operator theory , functional analysis , interpolation space , biochemistry , chemistry , genetics , repressor , biology , economic growth , political science , transcription factor , law , economics , gene
We present an extrapolation theorem for the convergence of sequences of Calderón‐Zygmund operators on various function spaces. We show that if a sequence of Calderón‐Zygmund operators with the same smoothness and decay parameters is convergent in operator norm on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2(\mathbb {R}^n)$\end{document} , then it is also convergent as operators on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^p(\mathbb {R}^n)$\end{document} , 1 < p < ∞, and on a large class of Triebel‐Lizorkin, Besov and Hardy spaces. As an application, we show that as the sampling density tends to the infinity, the wavelet frame operator tends to the identity in operator norm on Triebel‐Lizorkin, Besov and Hardy spaces.