Maximal operator on L p (·) spaces over spaces of homogeneous type: Local boundedness

Source abstract

Abstract We prove the local boundedness of the Hardy–Littlewood maximal operator on variable Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( X ) {L^{p(\,\cdot\,)}(X)} over spaces of homogeneous type for exponents p ⁢ ( ⋅ ) {p(\,\cdot\,)} which are bounded away from 1 and ∞ {\infty} and satisfy a local Muckenhoupt condition 𝒜 loc ⁢ ( X ) {\mathcal{A}_{\mathrm{loc}}(X)} . This generalizes a Euclidean theorem of A. K. Lerner. Our proof utilizes the machinery of adjacent dyadic grids.