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Potential theoretic approach to Schauder estimates for the fractional Laplacian


Authors: Claudia Bucur and Aram L. Karakhanyan
Journal: Proc. Amer. Math. Soc. 145 (2017), 637-651
MSC (2010): Primary 26A33, 35R11
DOI: https://doi.org/10.1090/proc/13227
Published electronically: July 26, 2016
MathSciNet review: 3577867
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Abstract: We present an elementary approach for the proof of Schauder estimates for the equation $ (-\Delta )^s u(x)=f(x), \,0<s<1$, with $ f$ having a modulus of continuity $ \omega _f$, based on the Poisson representation formula and dyadic ball approximation argument. We give the explicit modulus of continuity of $ u$ in balls $ B_r(x)\subset \mathbb{R}^n$ in terms of $ \omega _f$.


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Claudia Bucur
Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini, 50, 20100, Milano, Italy
Email: claudia.bucur@unimi.it

Aram L. Karakhanyan
Affiliation: Maxwell Institute for Mathematical Sciences and School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
Email: aram.karakhanyan@ed.ac.uk

DOI: https://doi.org/10.1090/proc/13227
Received by editor(s): February 15, 2016
Received by editor(s) in revised form: March 31, 2016
Published electronically: July 26, 2016
Additional Notes: The research of the second author was partially supported by an EPSRC grant
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society