We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1, 1). This result extends to corresponding maximal operators on a transitive group of isometrics of the tree, and in particular for (nonabelian finitely generated) free groups.
Cowling, M., Meda, S., Setti, A. (2010). A weak type $(1,1)$ estimate for a maximal operator on a group of isometries of homogeneous trees. COLLOQUIUM MATHEMATICUM, 118(1), 223-232 [10.4064/cm118-1-12].
A weak type $(1,1)$ estimate for a maximal operator on a group of isometries of homogeneous trees
MEDA, STEFANO;
2010
Abstract
We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1, 1). This result extends to corresponding maximal operators on a transitive group of isometrics of the tree, and in particular for (nonabelian finitely generated) free groups.
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