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

Characterization of function spaces via low regularity mollifiers

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

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.
Citation: Xavier Lamy, Petru Mironescu. Characterization of function spaces via low regularity mollifiers. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6015-6030. doi: 10.3934/dcds.2015.35.6015
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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), 81.

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H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Mathematical Library, (1978).

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H. Triebel, Theory of Function Spaces. II,, Monographs in Mathematics, (1992). doi: 10.1007/978-3-0346-0419-2.

show all references

References:
[1]

G. Bourdaud, Ondelettes et espaces de Besov,, Rev. Mat. Iberoamericana, 11 (1995), 477. doi: 10.4171/RMI/181.

[2]

L. Grafakos, Classical Fourier Analysis,, 2nd edition, (2008).

[3]

E. Stein, An $H^{1}$ function with nonsummable Fourier expansion,, in Harmonic Analysis (Cortona, (1982), 193. doi: 10.1007/BFb0069159.

[4]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, With the assistance of Timothy S. Murphy, (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), 81.

[6]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Mathematical Library, (1978).

[7]

H. Triebel, Theory of Function Spaces,, Monographs in Mathematics, (1983). doi: 10.1007/978-3-0346-0416-1.

[8]

H. Triebel, Theory of Function Spaces. II,, Monographs in Mathematics, (1992). doi: 10.1007/978-3-0346-0419-2.

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