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
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

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

[1]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020397

[2]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002

[3]

Yujuan Li, Huaifu Wang, Peipei Zhou, Guoshuang Zhang. Some properties of the cycle decomposition of WG-NLFSR. Advances in Mathematics of Communications, 2021, 15 (1) : 155-165. doi: 10.3934/amc.2020050

[4]

Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020389

[5]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285

[6]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[7]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[8]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[9]

Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146

[10]

P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178

[11]

Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048

[12]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[13]

Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300

[14]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[15]

Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322

[16]

Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1447-1468. doi: 10.3934/dcdsb.2020168

[17]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[18]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[19]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[20]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]