Wavelet decomposition of Calderón-Zygmund operators on function spaces

Abstract

We make use of the Beylkin-Coifman-Rokhlin wavelet decomposition algorithm on the CalderonZygmund kernel to obtain some fine estimates on the operator and prove the T(\) theorem on Besov and Triebel-Lizorkin spaces. This extends previous results of Frazier et at., and Han and Hofmann.2000 Mathematics subject classification: primary 42B20, 46B30.

An Approximation Method for Convolution Calderón-Zygmund Operators

“…These algorithms are also applicable to all Calderón-Zygmund operators and pseudodifferential operators. Since then, their algorithms are widely used in compression of matrices, operator approximation, and establishing boundedness of operators see [2][3][4][5][6][7][8][9]; In particular, Beylkin et al [1] approximated a class of Calderón-Zygmund operators by banded operators and gave the approximation accuracy. It is intriguing to know whether we can get some similar approximation methods on some more general spaces.…”

“…Let Q denote the collection of all dyadic cubes , , ∈ Z, ∈ Z . Now, we recall the following two definitions which can be found in [6,7,15].…”

“…for some fixed 1 , 2 > 0 independent of . For more details, we refer the reader to [6,15], This will play a key role in Section 3.…”

Boundedness of sublinear operators in Triebel–Lizorkin spaces via atoms

“…Finally, in Section 4, applying Theorem 2.1, we establish some criterion for boundedness of sublinear operators in Triebel-Lizorkin spaces, that is, we give the proof of Theorem 2.2. We point out that this criterion is useful in the study of boundedness for (sub)linear operators in Triebel-Lizorkin spaces; see, for example, [5,6,18].…”