Weak type estimates and Cotlar inequalities for Calder\'{o}n-Zygmund operators in nonhomogeneous spaces

TL;DR

This paper extends classical results about Calderón-Zygmund operators to nonhomogeneous spaces, establishing weak type bounds and Cotlar inequalities, including in abstract settings without Besicovich covering lemma.

Contribution

It proves that Calderón-Zygmund operators are of weak type in nonhomogeneous spaces and establishes Cotlar inequalities in both Euclidean and abstract settings.

Findings

Calderón-Zygmund operators are of weak type if bounded in L^2.

Cotlar's inequalities are established for maximal singular operators.

Weak type of maximal singular operator derived from these inequalities.

Abstract

In the paper we consider Calder\'{o}n-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calder\'{o}n-Zygmund operator is of weak type if it is bounded in $L^2$. We also prove several versions of Cotlar's inequality for maximal singular operator. One version of Cotlar's inequality (a simpler one) is proved in Euclidean setting, another one in a more abstract setting when Besicovich covering lemma is not available. We obtain also the weak type of maximal singular operator from these inequalities.