Convergence of sequences of Calderón‐Zygmund operators with application to wavelet expansions

Abstract

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.

A Jackson-type inequality associated with wavelet bases decomposition

“…Despite the feasibility of biframes, we paid the price of the linear independence, and we can deduce from Theorem 3.1 that the generator for the canonical dual bases requires mild decay to confirm the bijectivity of the frame operator on L p (R), 1 < p < ∞. Secondly, the conditions for Theorem 3.1 and Theorem 3.2 have been improved significantly in comparison with the results in [2,3,6], [27,Thm. 1.1] and [26,Thm.…”