Characterization of function spaces via low regularity mollifiers

Xavier  Lamy; Petru Mironescu

doi:10.3934/dcds.2015.35.6015

Article Contents

2015, Volume 35, Issue 12: 6015-6030. Doi: 10.3934/dcds.2015.35.6015

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Characterization of function spaces via low regularity mollifiers

Xavier Lamy^1, and
Petru Mironescu^2,

1.
Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex
2.
Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 november 1918, F-69622 Villeurbanne

Received: April 2014

Published: December 2015

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Abstract

Smoothness of a function $f:\mathbb{R}^n\to\mathbb{R}$ can be measured in terms of the rate of convergence of $f*\rho_{\epsilon}$ to $f$, where $\rho$ is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that $\rho$ is adapted to a given scale of spaces. Finally, we examine in detail the case where $\rho$ is a characteristic function.

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Mathematics Subject Classification: Primary: 46E35.

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References

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