Triangular maximal operators on locally finite trees

Abstract

We introduce the centred and the uncentred triangular maximal operators $$ and $$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $$ and $$ are bounded on $$ for every $$ in $$, that $$ is also bounded on $$, and that $$ is not of weak type (1, 1) on homogeneous trees. Our proof of the $$ boundedness of $$ hinges on the geometric approach of Córdoba and Fefferman. We also establish $$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy–Littlewood maximal operators (on balls) may be unbounded on $$ for every $$ even on some trees where the number of neighbours is uniformly bounded.