We prove the local boundedness of the Hardy–Littlewood maximal operator on variable Lebesgue spaces L p ( ⋅ ) ( X ) {Lp(\,·\,)(X)} over spaces of homogeneous type for exponents p ( ⋅ ) {p(\,·\,)} which are bounded away from 1 and ∞ {∞} and satisfy a local Muckenhoupt condition 𝒜 loc ( X ) {Aloc(X)} . This generalizes a Euclidean theorem of A. K. Lerner. Our proof utilizes the machinery of adjacent dyadic grids.
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Kopaliani et al. (2026) studied this question.
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