American Institute of Mathematical Sciences

Advanced
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, 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.  Google Scholar

[2]

L. Grafakos, Classical Fourier Analysis, 2nd edition, Graduate Texts in Mathematics, 249, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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,  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), 1, 1981, 81-97.  Google Scholar

[6]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam, 1978.  Google Scholar

[7]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[8]

H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 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), 477-512. doi: 10.4171/RMI/181.  Google Scholar

[2]

L. Grafakos, Classical Fourier Analysis, 2nd edition, Graduate Texts in Mathematics, 249, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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,  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), 1, 1981, 81-97.  Google Scholar

[6]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam, 1978.  Google Scholar

[7]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[8]

H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 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, 2021, 41 (7) : 3045-3062. doi: 10.3934/dcds.2020397
[2]

Radjesvarane Alexandre, Mouhamad Elsafadi. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II. Non cutoff case and non Maxwellian molecules. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 1-11. doi: 10.3934/dcds.2009.24.1
[3]

Yushi Nakano, Shota Sakamoto. Spectra of expanding maps on Besov spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1779-1797. doi: 10.3934/dcds.2019077
[4]

Baoxiang Wang. E-Besov spaces and dissipative equations. Communications on Pure & Applied Analysis, 2004, 3 (4) : 883-919. doi: 10.3934/cpaa.2004.3.883
[5]

Hermann Brunner, Jingtang Ma. Abstract cascading multigrid preconditioners in Besov spaces. Communications on Pure & Applied Analysis, 2006, 5 (2) : 349-365. doi: 10.3934/cpaa.2006.5.349
[6]

Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103
[7]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284
[8]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395
[9]

Houyu Jia, Xiaofeng Liu. Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces. Communications on Pure & Applied Analysis, 2008, 7 (4) : 845-852. doi: 10.3934/cpaa.2008.7.845
[10]

Mikil Foss, Joe Geisbauer. Higher differentiability in the context of Besov spaces for a class of nonlocal functionals. Evolution Equations & Control Theory, 2013, 2 (2) : 301-318. doi: 10.3934/eect.2013.2.301
[11]

Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967
[12]

Qing Hong, Guorong Hu. Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3103-3120. doi: 10.3934/cpaa.2019139
[13]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555
[14]

Julii A. Dubinskii. Complex Neumann type boundary problem and decomposition of Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 201-210. doi: 10.3934/dcds.2004.10.201
[15]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[16]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585
[17]

Hi Jun Choe, Bataa Lkhagvasuren, Minsuk Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2453-2464. doi: 10.3934/cpaa.2015.14.2453
[18]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078
[19]

Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
[20]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy