Pointwise estimates to the modified Riesz potential

Pointwise estimates to the modified Riesz potential In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev–Poincaré inequality. We show that this method gives a Sobolev–Poincaré inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev–Poincaré inequality depends on the domain. The Sobolev–Poincaré inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy–Littlewood maximal operator. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

Pointwise estimates to the modified Riesz potential

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-017-0983-y
Publisher site
See Article on Publisher Site

Abstract

In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev–Poincaré inequality. We show that this method gives a Sobolev–Poincaré inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev–Poincaré inequality depends on the domain. The Sobolev–Poincaré inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy–Littlewood maximal operator.

Journal

Manuscripta MathematicaSpringer Journals

Published: Nov 3, 2017

References

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