Triangular maximal operators on locally finite trees

Abstract

We introduce the centred and the uncentred triangular maximal operators $\mathcal T$ and $\mathcal U$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $\mathcal T$ and $\mathcal U$ are bounded on $L^p$ for every $p$ in $(1,\infty]$, that $\mathcal T$ is also bounded on $L^1(\mathfrak T)$, and that $\mathcal U$ is not of weak type $(1,1)$ on homogeneous trees. Our proof of the $L^p$ boundedness of $\mathcal U$ hinges on the geometric approach of A. C\'ordoba and R. Fefferman. We also establish $L^p$ 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 $L^p$ for every $p<\infty$ even on some trees where the number of neighbours is uniformly bounded.