#### Vol. 11, No. 2, 2018

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A sublinear version of Schur's lemma and elliptic PDE

### Stephen Quinn and Igor E. Verbitsky

Vol. 11 (2018), No. 2, 439–466
##### Abstract

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

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

$u-G\left({u}^{q}\sigma \right)=0,\phantom{\rule{1em}{0ex}}u\in {L}_{loc}^{q}\left(\Omega ,\sigma \right).$

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 .

##### Keywords
weighted norm inequalities, sublinear elliptic equations, Green's function, weak maximum principle, fractional Laplacian
##### Mathematical Subject Classification 2010
Primary: 35J61, 42B37
Secondary: 31B15, 42B25