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 $\Vert \nu \Vert =\nu (\Omega )$, and $G$ is an integral operator with nonnegative kernel, $\mathbf{G}\nu (x)={\int }_{\Omega }G(x,y)d\nu (y).$ These problems are motivated by sublinear elliptic equations in a domain $\Omega \subset {\mathbb{R}}^n$ 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 $M_{\alpha }$ ($0\le \alpha <n$) on ${\mathbb{R}}^n$, and characterize strong- and weak-type $(1,q)$-inequalities for $M_{\alpha }$ and more general maximal operators, as well as $(1,q)$-Carleson measure inequalities for Poisson integrals.

18 pages