Interior Schauder Estimates for Elliptic Equations Associated with Lévy Operators

We study the local regularity of solutions f to the integro-differential equation A f = g in U for open sets U ⊆ ℝ d , where A is the infinitesimal generator of a Lévy process ( X t ) t ≥ 0 . Under the assumption that the transition density of ( X t ) t ≥ 0 satisfies a certain gradient estimate, we...

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Bibliographic Details
Published in:Potential analysis 2022-03, Vol.56 (3), p.459-481
Main Author: Kühn, Franziska
Format: Article
Language:English
Subjects:
Differential equations
Elliptic functions
Estimates
Functional Analysis
Generators
Geometry
Mathematics
Mathematics and Statistics
Potential Theory
Probability Theory and Stochastic Processes
Stochastic processes
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Summary:We study the local regularity of solutions f to the integro-differential equation A f = g in U for open sets U ⊆ ℝ d , where A is the infinitesimal generator of a Lévy process ( X t ) t ≥ 0 . Under the assumption that the transition density of ( X t ) t ≥ 0 satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions f . Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.
ISSN:0926-2601
1572-929X
DOI:10.1007/s11118-020-09892-y