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

TL;DR

This paper investigates weighted norm inequalities of (1,q)-type for integral and fractional maximal operators, providing characterizations relevant to sublinear elliptic equations and harmonic analysis.

Contribution

It introduces new weighted inequality characterizations for integral and fractional maximal operators, extending understanding in elliptic PDEs and harmonic analysis.

Findings

Characterized strong- and weak-type (1,q) inequalities for integral operators.

Established (1,q)-Carleson measure inequalities for Poisson integrals.

Extended results to fractional maximal operators on n.

Abstract

We study weighted norm inequalities of $(1,q)$- type for $0<q<1$, $\Vert \mathbf{G} \nu \Vert_{L^q(\Omega, d \sigma)} \le C \, \Vert \nu \Vert, \quad \text{for all positive measures $\nu$ in $\Omega$},$ 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.