Vol. 1, No. 2, 2019

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Nonlocal self-improving properties: a functional analytic approach

Pascal Auscher, Simon Bortz, Moritz Egert and Olli Saari

Vol. 1 (2019), No. 2, 151–183
Abstract

A functional analytic approach to obtaining self-improving properties of solutions to linear nonlocal elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi–Mingione–Sire and Bass–Ren. Its flexibility is demonstrated by new applications to nonautonomous parabolic equations with nonlocal elliptic part and questions related to maximal regularity.

Keywords
Elliptic equations, fractional differentiability, nonlocal and stable-like operators, self-improving properties, analytic perturbation arguments, Cauchy problem for nonlocal parabolic equations
Mathematical Subject Classification 2010
Primary: 35D30, 35R11
Secondary: 26A33, 35K90, 46B70
Milestones
Received: 6 August 2017
Accepted: 15 December 2017
Published: 14 May 2018
Authors
Pascal Auscher
Laboratoire de Mathematique d’Orsay
Université de Paris-Sud, CNRS
Université Paris-Saclay
91405 Orsay
France
Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées
UMR 7352 du CNRS
Université de Picardie-Jules Verne
80039 Amiens
France
Simon Bortz
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
United States
Moritz Egert
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud, CNRS
Université Paris-Saclay
91405 Orsay
France
Olli Saari
Department of Mathematics and Systems Analysis
Aalto University
P.O. Box 11100
FI-00076 Aalto
Finland
Mathematical Institute
University of Bonn
53115 Bonn
Germany