A sublinear version of Schur's lemma and elliptic PDE

Abstract

We study the weighted norm inequality of $\left(1,q\right)$-type,

$$\parallel G\nu {\parallel }_{L^q\left(\Omega ,d\sigma \right)}\le C\parallel \nu \parallel \text{ for all }\nu \in {\mathcal{ℳ}}^{+}\left(\Omega \right),$$

along with its weak-type analogue, for $0<q<1$, where $G$ is an integral operator associated with the nonnegative kernel $G$ on $\Omega \times \Omega$. Here ${\mathcal{ℳ}}^{+}\left(\Omega \right)$ denotes the class of positive Radon measures in $\Omega$; $\sigma ,\nu \in {\mathcal{ℳ}}^{+}\left(\Omega \right)$, and $\parallel \nu \parallel =\nu \left(\Omega \right)$.

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 $\left(1,q\right)$-inequality for $0<q<1$ is closely connected with existence of a positive function $u$ such that $u\ge G\left(u^q\sigma \right)$, i.e., a supersolution to the integral equation

$$u-G\left(u^q\sigma \right)=0,u\in L_{loc}^q\left(\Omega ,\sigma \right).$$

This study is motivated by solving sublinear equations involving the fractional Laplacian,

$${\left(-\Delta \right)}^{\frac{\alpha }{2}}u-u^q\sigma =0,$$

in domains $\Omega \subseteq {ℝ}^n$ which have a positive Green function $G$ for $0<\alpha <n$.