A necessary condition for Calder�n-Zygmund singular integral operators

Abstract

ABSTRACT. CaIder6n-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderdn-Zygmund convolution singular integral operator is bounded on the Hardy space H 1 (Rn), then the homogeneous of degree zero kernel is in the Hardy space HI (s n-l) on the sphere.

Classical Fourier Analysis

“…This result is sharp in the sense that for such Ω , T Ω may not map H 1 (R n ) to L 1,q (R n ) when q < 2 in general. If T Ω maps H 1 (R n ) to itself, Daly and Phillips [87] (in dimension n = 2) and Daly [86] (in dimensions n ≥ 3) showed that Ω must lie in the Hardy space H 1 (S n−1 ). There are also results concerning the singular maximal operator M Ω ( f )(x) = sup r>0 1 vnr n |y|≤r | f (x − y)||Ω (y)| dy, where Ω is an integrable function on S n−1 of not necessarily vanishing integral.…”