WEAK TYPE ESTIMATES AND COTLAR INEQUALITIES FOR CALDERÓN-ZYGMUND OPERATORS IN NONHOMOGENEOUS

SPACES

FEDOR NAZAROV SERGEI TREIL ALEXANDER VOLBERG

Abstract. 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.