Abstract

Abstract. Any sufficiently smooth one-dimensional Calderón-Zygmund convolution op-erator is the average of Haar shift operators. The latter are dyadic operators which can be efficiently expressed in terms of the Haar basis. This extends the result of S. Petermichl [11] on restoring Hilbert transform via Haar shift operators, a technique that has become fundamental to the analysis of these operators. 1. Introduction. We will represent one-dimensional Calderón-Zygmund convolution operators with suf-ficiently smooth kernels, see (2.5), (2.6), by means of a properly chosen averaging of certain Haar shift operators with bounded coefficients. By Haar shift operators, we mean for ex-ample (2.9): linear operators that can be expressed in an efficient manner with the Haar