Improved Buckley’s theorem on locally compact abelian groups

Abstract

We present sharp quantitative weighted norm inequalities for the Hardy-Littlewood maximal function in the context of Locally Compact Abelian Groups, obtaining an improved version of the so-called Buckley's Theorem. On the way, we prove a precise reverse Hölder inequality for Muckenhoupt A∞ weights and provide a valid version of the "open property" for Muckenhoupt Ap weights.1991 Mathematics Subject Classification. Primary: 42B25. Secondary: 43A70.

Maximal operators on the infinite-dimensional torus

“…As a first positive result in the context of R, we note that the local Calderón-Zygmund decomposition allows one to prove that the class of A R ∞ (T ω ) weights is inside the class of reverse Hölder weights. Moreover, we can prove a sharp reverse Hölder inequality in the spirit of those in [HPR12,PR19] but in the sharper form obtained in [PR18] adapted to flat weights. This follows from standard arguments and the fact that the cubes in the local Calderón-Zygmund decomposition are cubes of R. .…”

“…194], there are no D -sequences in T ω , the infinitedimensional torus. Hence, we lack the existence of the base sets considered in [PR19] and in particular, we lack the automatic validity of the Lebesgue differentiation theorem. Even in case we are able to find a different base of sets to work with, we may still lack the corresponding version of Lebesgue differentiation theorem.…”