DOI: 10.1515/gmj-2026-3048 ISSN: 1072-947X

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

Tengiz Kopaliani, Alina Shalukhina, George Tephnadze
Show PDF Cite

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.

More from our Archive