Weak type $(1,1)$ estimates of maximal truncated singular operators

Abstract

Let $X$ be a space of homogeneous type and $T$ a singular integral operator which is bounded on $L^2(X)$. We give a sufficient condition on the kernel of $T$ so that the maximal truncated operator $T_*$, which is defined by $T_*f(x) = sup_{\epsilon\lt0} |T_\epsilonf(x)|$, to be of weak type (1, 1). Our condition is weaker than the usual Hörmander type condition. Applications include the dominated convergence theorem of holomorphic functional calculi of linear elliptic operators on irregular domains.