Weighted norm inequalities of (1,q)-type for integral and fractional maximal operators

Abstract

We study weighted norm inequalities of $(1,q)$- type for $0<q<1$, $\Vert \mathbf{G} ν\Vert_{L^q(Ω, d σ)} \le C \, \Vert ν\Vert, \quad \text{for all positive measures $ν$ in $Ω$},$ along with their weak-type counterparts, where $\Vert ν\Vert=ν(Ω)$, and $G$ is an integral operator with nonnegative kernel, $\mathbf{G} ν(x) = \int_ΩG(x, y) d ν(y).$ These problems are motivated by sublinear elliptic equations in a domain $Ω\subset\mathbb{R}^n$ with non-trivial Green's function $G(x, y)$ associated with the Laplacian, fractional Laplacian, or more general elliptic operator. We also treat fractional maximal operators $M_α$ ($0\le α<n$) on $\mathbb{R}^n$, and characterize strong- and weak-type $(1,q)$-inequalities for $M_α$ and more general maximal operators, as well as $(1,q)$-Carleson measure inequalities for Poisson integrals.