Analysis & PDE

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

A sublinear version of Schur's lemma and elliptic PDE

Stephen Quinn and Igor E. Verbitsky

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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.

Article information

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J61: Semilinear elliptic equations 42B37: Harmonic analysis and PDE [See also 35-XX]
Secondary: 31B15: Potentials and capacities, extremal length 42B25: Maximal functions, Littlewood-Paley theory

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


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.

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