Local maximal function and weights in a general setting

Abstract

For a proper open set $\Omega$ immersed in a metric space with the weak homogeneity property, and given a measure $\mu$ doubling on a certain family of balls lying “well inside” of $\Omega$, we introduce a local maximal function and characterize the weights $w$ for which it is bounded on $L^p(\Omega ,w d\mu )$ when $1<p<\infty$ and of weak type $(1,1)$. We generalize previous known results and we also present an application to interior Sobolev’s type estimates for appropriate solutions of the differential equation $\Delta ^m u=f$, satisfied in an open proper subset $\Omega$ of $\mathbb R ^n$. Here, the data $f$ belongs to some weighted $L^p$ space that could allow functions to increase polynomially when approaching the boundary of $\Omega$.