Pseudo-localisation of singular integrals in $L^p$

Abstract

As a step in developing a non-commutative Calderón-Zygmund theory, J. Parcet (J. Funct. Anal. {\bf 256} (2009), no. 2, 509-593) 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 Calderón-Zygmund operator $T$. Parcet also asked if a similar result holds true in $L^p$ for $p\in(1,\infty)$. 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.