A sublinear version of Schur's lemma and elliptic PDE

Abstract

We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{G}ν\Vert_{L^q(Ω, dσ)} \le C \Vert ν\Vert, \quad \text{ for all } ν\in \mathcal{M}^+(Ω), \] 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 $\mathcal{M}^+(Ω)$ denotes the class of positive Radon measures in $Ω$; $σ, ν\in \mathcal{M}^+(Ω)$, and $||ν||=ν(Ω)$. For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $σ$ 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 $u \ge \mathbf{G}(u^q σ)$, i.e., a supersolution to the integral equation \[ u - \mathbf{G}(u^q σ) = 0, \quad u \in L^q_{\rm loc} (Ω, σ). \] This study is motivated by solving sublinear equations involving the fractional Laplacian, \[ (-Δ)^{\fracα{2}} u - u^q σ= 0\] in domains $Ω\subseteq \mathbf{R}^n$ which have a positive Green function $G$, for $0 < α< n$.