I'm looking for references discussing maximal (strong and weak) inequalities for doubling measure spaces. More precisely, Let $(X,\mu, d)$ be a metric measure space with $\mu$ being a doubling measure: What is known for the following maximal function: $$ Mf(x)=\sup_{r>0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|f|d\mu$$
2026-03-29 20:54:14.1774817654
Hardy-littlewood maximal inequality for doubling measures
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There is an article written by Vagif S. Guliyev and Stefan G. Samko called "Maximal Operator in Variable Exponent Generalized Morrey Spaces on Quasi-metric Measure Space". In this paper, they prove theorems on the boundedness of the maximal operator that you mentioned.