Pseudo-localisation of singular integrals in L^p

TL;DR

This paper extends Parcet's pseudo-localisation principle for classical singular integrals from L^2 to all L^p spaces with 1<p<infinity, using martingale techniques and improving the original results.

Contribution

It proves a pseudo-localisation principle for singular integrals in L^p spaces, generalizing and strengthening Parcet's earlier L^2 result.

Findings

Pseudo-localisation holds in L^p for 1<p<infinity.

The proof uses martingale techniques.

The result improves on the original L^2 case.

Abstract

As a step in developing a non-commutative Calderon-Zygmund theory, J. Parcet (J. Funct. Anal., 2009) established a new pseudo-localisation principle for classical singular integrals, showing that Tf has small L^2 norm outside a set which only depends on f in L^2 but not on the arbitrary normalised Calderon-Zygmund operator T. Parcet also asked if a similar result holds true in L^p for 1 < p < infinity. This is answered in the affirmative in the present paper. The proof, which is based on martingale techniques, even somewhat improves on the original L^2 result.