Revista Matemática Iberoamericana

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Volume 27, Issue 2, 2011, pp. 557–584
DOI: 10.4171/RMI/646

Published online: 2011-08-31

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

Tuomas Hytönen[1]

(1) University of Helsinki, Finland

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.

Keywords: Calderón–Zygmund operator, T (1) theorem, operations on the Haar basis

Hytönen Tuomas: Pseudo-localisation of singular integrals in $L^p$. Rev. Mat. Iberoam. 27 (2011), 557-584. doi: 10.4171/RMI/646