Analysis & PDE
- Anal. PDE
- Volume 11, Number 2 (2018), 439-466.
A sublinear version of Schur's lemma and elliptic PDE
We study the weighted norm inequality of -type,
along with its weak-type analogue, for , where is an integral operator associated with the nonnegative kernel on . Here denotes the class of positive Radon measures in ; , and .
For both weak-type and strong-type inequalities, we provide conditions which characterize the measures for which such an embedding holds. The strong-type -inequality for is closely connected with existence of a positive function such that , i.e., a supersolution to the integral equation
This study is motivated by solving sublinear equations involving the fractional Laplacian,
in domains which have a positive Green function for .
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
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