Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators in nonhomogeneous spaces
Fedor Nazarov, Sergei Treil, and Alexander Volberg
In the paper we consider Calderón–Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calderón–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.