Characterization of function spaces via low regularity mollifiers

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December 2015, 35(12): 6015-6030. doi: 10.3934/dcds.2015.35.6015

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, France
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 May 2015

Abstract
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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.

Keywords: Littlewood-Paley decomposition., approximation, mollifiers, Besov spaces.

Mathematics Subject Classification: Primary: 46E3.

Citation: Xavier Lamy, Petru Mironescu. Characterization of function spaces via low regularity mollifiers. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6015-6030. doi: 10.3934/dcds.2015.35.6015

References:

[1]	G. Bourdaud, Ondelettes et espaces de Besov, Rev. Mat. Iberoamericana, 11 (1995), 477-512. doi: 10.4171/RMI/181.
[2]	L. Grafakos, Classical Fourier Analysis, 2nd edition, Graduate Texts in Mathematics, 249, Springer, New York, 2008.
[3]	E. Stein, An $H^{1}$ function with nonsummable Fourier expansion, in Harmonic Analysis (Cortona, 1982), Lecture Notes in Math., 992, Springer, Berlin, 1983, 193-200. doi: 10.1007/BFb0069159.
[4]	E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993,
[5]	E. Stein, M. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain $H^p$ classes, in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1, 1981, 81-97.
[6]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam, 1978.
[7]	H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[8]	H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.

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References:

[1]	G. Bourdaud, Ondelettes et espaces de Besov, Rev. Mat. Iberoamericana, 11 (1995), 477-512. doi: 10.4171/RMI/181.
[2]	L. Grafakos, Classical Fourier Analysis, 2nd edition, Graduate Texts in Mathematics, 249, Springer, New York, 2008.
[3]	E. Stein, An $H^{1}$ function with nonsummable Fourier expansion, in Harmonic Analysis (Cortona, 1982), Lecture Notes in Math., 992, Springer, Berlin, 1983, 193-200. doi: 10.1007/BFb0069159.
[4]	E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993,
[5]	E. Stein, M. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain $H^p$ classes, in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1, 1981, 81-97.
[6]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam, 1978.
[7]	H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[8]	H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.

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