Abstract: In this talk we consider the centred and the uncentred Hardy--Littlewood maximal operators -- denoted $\cM$ and $\cN$, respectively -- on certain metric measure spaces with exponential volume growth and ``locally bounded geometry'' such as the hyperbolic upper half-plane and homogeneous trees.
A classical result of Hardy and Littlewood states that on the Euclidean space both $\cM$ and $\cN$ are bounded on $L^p$ for all $p>1$ and that they are of weak type $(1,1)$.
We discuss the problem of extending this result to the hyperbolic upper half-plane, to homogeneous trees and to more general metric measure spaces, including, for instance, non-homogeneous trees and Cartan--Hadamard manifolds with pinched negative curvature.
In particular, we shall:
(i) illustrate the fact that even on the hyperbolic upper-half space~$\cM$ and $\cN$ are non-equivalent operators and have different $L^p$ boundedness properties;
(ii) analyse the $L^p$ boundedness properties of $\cM$ and $\cN$ on certain examples of non-homogeneous trees and Cartan--Hadamard manifolds with pinched negative curvature;
(iii) show how the results in (ii) are consequences of a more general result on a recently introduced class of Gromov hyperbolic spaces, called spider's webs;
(iv) introduce a variant of $\cN$ in which metric balls are replaced by ``half-balls'' and illustrate the corresponding $L^p$ mapping properties, which are, perhaps surprisingly, quite different from those of $\cN$.
This is joint work with Effie Papageorgiou (Paderborn), Federico Santagati (Politecnico di Torino) and Nikos Chalmoukis (Milano--Bicocca).
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