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 (1,q)-type,

GνLq(Ω,dσ) Cν 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 u G(uqσ), i.e., a supersolution to the integral equation

u G(uqσ) = 0,u L locq(Ω,σ).

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

(Δ)α 2 u uqσ = 0,

in domains Ω n which have a positive Green function G for 0 < α < n.

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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
Milestones
Received: 10 February 2017
Revised: 14 July 2017
Accepted: 5 September 2017
Published: 17 October 2017
Authors
Stephen Quinn
Department of Mathematics
University of Missouri
Columbia, MO
United States
Igor E. Verbitsky
Department of Mathematics
University of Missouri
Columbia, MO
United States