Triangular maximal operators on locally finite trees

We introduce the centred and the uncentred triangular maximal operators mathematical equation and mathematical equation, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both mathematical equation and mathematical equation are bounded on mathematical equation for every mathematical equation in mathematical equation, that mathematical equation is also bounded on mathematical equation, and that mathematical equation is not of weak type (1, 1) on homogeneous trees. Our proof of the mathematical equation boundedness of mathematical equation hinges on the geometric approach of Córdoba and Fefferman. We also establish mathematical equation 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 mathematical equation for every mathematical equation even on some trees where the number of neighbours is uniformly bounded.