# A sublinear version of Schur's lemma and elliptic PDE

## TL;DR

This paper characterizes weighted norm inequalities of sublinear integral operators related to the fractional Laplacian, providing conditions for their validity and connecting them to solutions of nonlinear PDEs involving fractional powers of the Laplacian.

## Contribution

It introduces new criteria for weighted inequalities of sublinear operators for q<1 and links these to existence of supersolutions for fractional PDEs.

## Key findings

- Characterization of measures for which the weighted inequality holds.
- Connection between strong-type inequalities and supersolutions of integral equations.
- Application to sublinear equations involving the fractional Laplacian.

## Abstract

We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{G}\nu \Vert_{L^q(\Omega, d\sigma)} \le C \Vert \nu \Vert, \quad \text{ for all } \nu \in \mathcal{M}^+(\Omega), \] along with its weak-type analogue, for $0 < q < 1$, where $\mathbf{G}$ is an integral operator associated with the nonnegative kernel $G(x,y)$. Here $\mathcal{M}^+(\Omega)$ denotes the class of positive Radon measures in $\Omega$; $\sigma, \nu \in \mathcal{M}^+(\Omega)$, and $||\nu||=\nu(\Omega)$. For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $\sigma$ for which such an embedding holds. The strong-type $(1,q)$-inequality for $0<q<1$ is closely connected with existence of a positive function $u$ such that $u \ge \mathbf{G}(u^q \sigma)$, i.e., a supersolution to the integral equation \[ u - \mathbf{G}(u^q \sigma) = 0, \quad u \in L^q_{\rm loc} (\Omega, \sigma). \] This study is motivated by solving sublinear equations involving the fractional Laplacian, \[ (-\Delta)^{\frac{\alpha}{2}} u - u^q \sigma = 0\] in domains $\Omega \subseteq \mathbf{R}^n$ which have a positive Green function $G$, for $0 < \alpha < n$.