On commutators of singular integrals and bilinear singular integrals

Abstract

Lp estimates for multilinear singular integrals generalizing Caldern's commutator integral are obtained.The methods introduced involve Fourier and Mellin analysis.1.In this paper we introduce new methods to obtain estimates for commutators of singular integrals as well as other related operators.Let A(f) = A(x)f(x) and H(f) = pvff(t)dt/(x -t).It has been shown byCaldern's theorem was proved using a characterization of the Hardy space HX(R) in terms of the Lusin area function.It is the special form of the kernel which permits analysis using complex variables.It will be shown here that these V estimates (as well as others) can also be obtained by making use of the Fourier transform followed by the Mellin transform.These ideas are applied to obtain estimates of the following typewhere l/r -l/px + l/p2 + l/p3, 1 < px < , 1 < p2 < , 1 < p3 < .Actually, results improving both (1.1) (1.2) can be obtained.In fact let a" h r\M -" f"y)dy, J- (x-y)3 where a = dA/dx, b = dB/dx.Then for > p > 0 we have (1.3) f\C(a, b, f) (x)\p dx < Cp f(a*(x)b*(x)f*(x)f dx, where a* is the Hardy-Littlewood maximal function of a.