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2018 A sublinear version of Schur's lemma and elliptic PDE
Stephen Quinn, Igor E. Verbitsky
Anal. PDE 11(2): 439-466 (2018). DOI: 10.2140/apde.2018.11.439

Abstract

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

G ν L q ( Ω , d σ ) C ν  for all  ν + ( Ω ) ,

along with its weak-type analogue, for 0<q<1, where G is an integral operator associated with the nonnegative kernel G on Ω×Ω. Here +(Ω) denotes the class of positive Radon measures in Ω; σ,ν+(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 uG(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 Ωn which have a positive Green function G for 0<α<n.

Citation

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Stephen Quinn. Igor E. Verbitsky. "A sublinear version of Schur's lemma and elliptic PDE." Anal. PDE 11 (2) 439 - 466, 2018. https://doi.org/10.2140/apde.2018.11.439

Information

Received: 10 February 2017; Revised: 14 July 2017; Accepted: 5 September 2017; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 1384.35031
MathSciNet: MR3724493
Digital Object Identifier: 10.2140/apde.2018.11.439

Subjects:
Primary: 35J61 , 42B37
Secondary: 31B15 , 42B25

Keywords: fractional Laplacian , Green's function , sublinear elliptic equations , weak maximum principle , weighted norm inequalities

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.11 • No. 2 • 2018
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