Geodesic
Weighted norm inequalities for the maximal operator on L^p(·) over spaces of homogeneous type
David Cruz-Uribe, OFS1 ; Jeremy Cummings1
1The University of Alabama, Department of Mathematics
Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 457-488

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Given a space of homogeneous type $(X,d,\mu)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^{p(\cdot)}$. We prove that the variable Muckenhoupt condition $A_{p(\cdot)}$ is necessary and sufficient for the strong type inequality if $p(\cdot)$ satisfies log-Hölder continuity conditions and $1 < p_- \leq p_+ < \infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved by Cruz-Uribe, Fiorenza and Neugebauer (2012).  
DOI : 10.54330/afm.115059
Keywords: Variable Lebesgue spaces, maximal operator, two weights, spaces of homogeneous type

David Cruz-Uribe, OFS  1   ; Jeremy Cummings  1

1 The University of Alabama, Department of Mathematics 
David Cruz-Uribe, OFS; Jeremy Cummings. Weighted norm inequalities for the maximal operator on L^p(·) over spaces of homogeneous type. Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 457-488. doi: 10.54330/afm.115059
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