## Analysis & PDE

• Anal. PDE
• Volume 11, Number 2 (2018), 439-466.

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

#### Abstract

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

along with its weak-type analogue, for $0, where $G$ is an integral operator associated with the nonnegative kernel $G$ on $Ω×Ω$. Here $ℳ+(Ω)$ denotes the class of positive Radon measures in $Ω$; $σ,ν∈ℳ+(Ω)$, 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 is closely connected with existence of a positive function $u$ such that $u≥G(uqσ)$, i.e., a supersolution to the integral equation

$u − G ( u q σ ) = 0 , u ∈ L l o c q ( Ω , σ ) .$

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

$( − Δ ) α 2 u − u q σ = 0 ,$

in domains $Ω⊆ℝn$ which have a positive Green function $G$ for $0<α.

#### Article information

Source
Anal. PDE, Volume 11, Number 2 (2018), 439-466.

Dates
Revised: 14 July 2017
Accepted: 5 September 2017
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513774510

Digital Object Identifier
doi:10.2140/apde.2018.11.439

Mathematical Reviews number (MathSciNet)
MR3724493

Zentralblatt MATH identifier
1384.35031

#### Citation

Quinn, Stephen; Verbitsky, Igor E. A sublinear version of Schur's lemma and elliptic PDE. Anal. PDE 11 (2018), no. 2, 439--466. doi:10.2140/apde.2018.11.439. https://projecteuclid.org/euclid.apde/1513774510

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