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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

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 & Continuous Dynamical Systems - A, 2015, 35 (12) : 6015-6030. doi: 10.3934/dcds.2015.35.6015
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show all references

References:
[1]

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

[2]

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

[3]

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

[4]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, With the assistance of Timothy S. Murphy, (1993).   Google Scholar

[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.   Google Scholar

[6]

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

[7]

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

[8]

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

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