American Institute of Mathematical Sciences

Advanced
January  2013, 12(1): 59-88. doi: 10.3934/cpaa.2013.12.59

Whitney-type extensions in quasi-metric spaces

Ryan Alvarado 1, , Irina Mitrea 2, and  Marius Mitrea 3,

1. 

Department of Mathematics, University of Missouri at Columbia, Columbia, MO 65211, United States
2. 

Department of Mathematics, Temple University, Philadelphia, PA 19122, United States
3. 

Department of Mathematics, University of Missouri, Columbia, MO 65211

Received  May 2011 Revised  March 2012 Published  September 2012

We discuss geometrical scenarios guaranteeing that functions defined on a given set may be extended to the entire ambient, with preservation of the class of regularity. This extends to arbitrary quasi-metric spaces work done by E.J. McShane in the context of metric spaces, and to geometrically doubling quasi-metric spaces work done by H. Whitney in the Euclidean setting. These generalizations are quantitatively sharp.
Keywords: quantitative Urysohn lemma, extension of Hölder functions, extension of Lipschitz functions, Whitney decomposition., geometrically doubling quasi-metric space, partition of unity, quasi-metric space, Hölder functions, metrization, Quasi-metric space, Lipschitz functions, Whitney extension.
Mathematics Subject Classification: Primary: 26A16, 26B35; Secondary 26B05, 54E3.
Citation: Ryan Alvarado, Irina Mitrea, Marius Mitrea. Whitney-type extensions in quasi-metric spaces. Communications on Pure & Applied Analysis, 2013, 12 (1) : 59-88. doi: 10.3934/cpaa.2013.12.59
References:
[1]

T. Aoki, Locally bounded linear topological spaces,, Proc. Japan Acad. Tokyo, 18 (1942), 588.  doi: 10.3792/pia/1195573733.  Google Scholar

[2]

Y. Brudnyi and P. Shvartsman, Whitney's extension problem for multivariate $C^{1,\omega}$-functions,, Trans. Amer. Math. Soc., 353 (2001), 2487.  doi: 10.1090/S0002-9947-01-02756-8.  Google Scholar

[3]

L. Chen, Smoothness and smooth extensions (I): Generalization of MWK functions and gradually varied functions,, preprint, ().   Google Scholar

[4]

L. Chen, A digital-discrete method for smooth-continuous data reconstruction,, preprint, ().   Google Scholar

[5]

R. R. Coifman and G. Weiss, "Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes,", Lecture Notes in Mathematics, Vol. 242 (1971).   Google Scholar

[6]

R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis,, Bull. Amer. Math. Soc., 83 (1977), 569.  doi: 10.1090/S0002-9904-1977-14325-5.  Google Scholar

[7]

R. Engelking, "General Topology,", Heldermann Verlag, (1989).   Google Scholar

[8]

C. Fefferman and B. Klartag, An example related to Whitney extension with almost minimal $C^m$ norm,, Rev. Mat. Iberoamericana, 25 (2009), 423.  doi: 10.4171/RMI/571.  Google Scholar

[9]

C. Fefferman, A sharp form of Whitney's extension theorem,, Ann. of Math., 161 (2005), 509.  doi: 10.4007/annals.2005.161.509.  Google Scholar

[10]

H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969).   Google Scholar

[11]

A. Gogatishvili, P. Koskela and N. Shanmugalingam, Interpolation properties of Besov spaces defined on metric spaces,, Mathematische Nachrichten, Vol. 283 (2010), 215.  doi: 10.1002/mana.200810242.  Google Scholar

[12]

J. Heinonen, "Lectures on Analysis on Metric Spaces,", Springer-Verlag, (2001).  doi: 10.1007/978-1-4613-0131-8.  Google Scholar

[13]

E. Hille and R. S. Phillips, "Functional Analysis and Semigroups,", Amer. Math. Soc. Colloq. Publ., Vol. 31 (1957).   Google Scholar

[14]

L. Hörmander, On the division of distributions by polynomials,, Ark. Mat., 3 (1958), 555.  doi: 10.1007/BF02589517.  Google Scholar

[15]

A. Jonsson and H. Wallin, "Function Spaces on Subsets of $R^n$,", Math. Rep., Vol. 2 (1984).   Google Scholar

[16]

N. J. Kalton, N. T. Peck and J. W. Roberts, "An $F$-space Sampler,", London Math. Society Lecture Notes Series, (1984).   Google Scholar

[17]

M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche transformationen,, Fundamenta Mathematicae, 22 (1934), 77.   Google Scholar

[18]

P. Koskela, N. Shanmugalingam and H. Tuominen, Removable sets for the Poincaré inequality on metric spaces,, Indiana Math. J., 49 (2000), 333.  doi: 10.1512/iumj.2000.49.1719.  Google Scholar

[19]

S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory,, Expositiones Mathematicae, 3 (1983), 193.   Google Scholar

[20]

R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 257.  doi: 10.1016/0001-8708(79)90012-4.  Google Scholar

[21]

R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 271.  doi: 10.1016/0001-8708(79)90013-6.  Google Scholar

[22]

E. J. McShane, Extension of range of functions,, Bull. Amer. Math. Soc., 40 (1934), 837.  doi: 10.1090/S0002-9904-1934-05978-0.  Google Scholar

[23]

D. Mitrea, I. Mitrea, M. Mitrea and S. Monniaux, "Groupoid Metrization Theory with Applications to Analysis on Quasi-Metric Spaces and Functional Analysis,", to appear in the Applied and Numerical Harmonic Analysis monograph series, (2012).   Google Scholar

[24]

J. R. Munkres, "Topology,", Second edition, (2000).   Google Scholar

[25]

S. Rolewicz, On a certain class of linear metric spaces,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 5 (1957), 471.   Google Scholar

[26]

R. A. Rosenbaum, Subadditive functions,, Duke Math. J., 17 (1950), 227.  doi: 10.1215/S0012-7094-50-01721-2.  Google Scholar

[27]

A. Rosenfeld, Continuous functions on digital pictures,, Pattern Recognition Letters, 4 (1986), 177.  doi: 10.1016/0167-8655(86)90017-6.  Google Scholar

[28]

J. T. Schwartz, "Nonlinear Functional Analysis,", Gordon and Breach Science Publishers, (1969).   Google Scholar

[29]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).   Google Scholar

[30]

F. A. Valentine, On the extension of a vector function so as to preserve a Lipschitz condition,, Bulletin of the American Mathematical Society, 49 (1943), 100.  doi: 10.1090/S0002-9904-1943-07859-7.  Google Scholar

[31]

F. A. Valentine, A Lipschitz condition preserving extension for a vector function,, American Journal of Mathematics, 67 (1945), 83.  doi: 10.2307/2371917.  Google Scholar

[32]

H. Whitney, Analytic extensions of functions defined on closed sets,, Trans. Amer. Math. Soc., 36 (1934), 63.  doi: 10.1090/S0002-9947-1934-1501735-3.  Google Scholar

show all references

References:
[1]

T. Aoki, Locally bounded linear topological spaces,, Proc. Japan Acad. Tokyo, 18 (1942), 588.  doi: 10.3792/pia/1195573733.  Google Scholar

[2]

Y. Brudnyi and P. Shvartsman, Whitney's extension problem for multivariate $C^{1,\omega}$-functions,, Trans. Amer. Math. Soc., 353 (2001), 2487.  doi: 10.1090/S0002-9947-01-02756-8.  Google Scholar

[3]

L. Chen, Smoothness and smooth extensions (I): Generalization of MWK functions and gradually varied functions,, preprint, ().   Google Scholar

[4]

L. Chen, A digital-discrete method for smooth-continuous data reconstruction,, preprint, ().   Google Scholar

[5]

R. R. Coifman and G. Weiss, "Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes,", Lecture Notes in Mathematics, Vol. 242 (1971).   Google Scholar

[6]

R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis,, Bull. Amer. Math. Soc., 83 (1977), 569.  doi: 10.1090/S0002-9904-1977-14325-5.  Google Scholar

[7]

R. Engelking, "General Topology,", Heldermann Verlag, (1989).   Google Scholar

[8]

C. Fefferman and B. Klartag, An example related to Whitney extension with almost minimal $C^m$ norm,, Rev. Mat. Iberoamericana, 25 (2009), 423.  doi: 10.4171/RMI/571.  Google Scholar

[9]

C. Fefferman, A sharp form of Whitney's extension theorem,, Ann. of Math., 161 (2005), 509.  doi: 10.4007/annals.2005.161.509.  Google Scholar

[10]

H. Federer, "Geometric Measure Theory,", Springer-Verlag, (1969).   Google Scholar

[11]

A. Gogatishvili, P. Koskela and N. Shanmugalingam, Interpolation properties of Besov spaces defined on metric spaces,, Mathematische Nachrichten, Vol. 283 (2010), 215.  doi: 10.1002/mana.200810242.  Google Scholar

[12]

J. Heinonen, "Lectures on Analysis on Metric Spaces,", Springer-Verlag, (2001).  doi: 10.1007/978-1-4613-0131-8.  Google Scholar

[13]

E. Hille and R. S. Phillips, "Functional Analysis and Semigroups,", Amer. Math. Soc. Colloq. Publ., Vol. 31 (1957).   Google Scholar

[14]

L. Hörmander, On the division of distributions by polynomials,, Ark. Mat., 3 (1958), 555.  doi: 10.1007/BF02589517.  Google Scholar

[15]

A. Jonsson and H. Wallin, "Function Spaces on Subsets of $R^n$,", Math. Rep., Vol. 2 (1984).   Google Scholar

[16]

N. J. Kalton, N. T. Peck and J. W. Roberts, "An $F$-space Sampler,", London Math. Society Lecture Notes Series, (1984).   Google Scholar

[17]

M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche transformationen,, Fundamenta Mathematicae, 22 (1934), 77.   Google Scholar

[18]

P. Koskela, N. Shanmugalingam and H. Tuominen, Removable sets for the Poincaré inequality on metric spaces,, Indiana Math. J., 49 (2000), 333.  doi: 10.1512/iumj.2000.49.1719.  Google Scholar

[19]

S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory,, Expositiones Mathematicae, 3 (1983), 193.   Google Scholar

[20]

R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 257.  doi: 10.1016/0001-8708(79)90012-4.  Google Scholar

[21]

R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type,, Adv. in Math., 33 (1979), 271.  doi: 10.1016/0001-8708(79)90013-6.  Google Scholar

[22]

E. J. McShane, Extension of range of functions,, Bull. Amer. Math. Soc., 40 (1934), 837.  doi: 10.1090/S0002-9904-1934-05978-0.  Google Scholar

[23]

D. Mitrea, I. Mitrea, M. Mitrea and S. Monniaux, "Groupoid Metrization Theory with Applications to Analysis on Quasi-Metric Spaces and Functional Analysis,", to appear in the Applied and Numerical Harmonic Analysis monograph series, (2012).   Google Scholar

[24]

J. R. Munkres, "Topology,", Second edition, (2000).   Google Scholar

[25]

S. Rolewicz, On a certain class of linear metric spaces,, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 5 (1957), 471.   Google Scholar

[26]

R. A. Rosenbaum, Subadditive functions,, Duke Math. J., 17 (1950), 227.  doi: 10.1215/S0012-7094-50-01721-2.  Google Scholar

[27]

A. Rosenfeld, Continuous functions on digital pictures,, Pattern Recognition Letters, 4 (1986), 177.  doi: 10.1016/0167-8655(86)90017-6.  Google Scholar

[28]

J. T. Schwartz, "Nonlinear Functional Analysis,", Gordon and Breach Science Publishers, (1969).   Google Scholar

[29]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).   Google Scholar

[30]

F. A. Valentine, On the extension of a vector function so as to preserve a Lipschitz condition,, Bulletin of the American Mathematical Society, 49 (1943), 100.  doi: 10.1090/S0002-9904-1943-07859-7.  Google Scholar

[31]

F. A. Valentine, A Lipschitz condition preserving extension for a vector function,, American Journal of Mathematics, 67 (1945), 83.  doi: 10.2307/2371917.  Google Scholar

[32]

H. Whitney, Analytic extensions of functions defined on closed sets,, Trans. Amer. Math. Soc., 36 (1934), 63.  doi: 10.1090/S0002-9947-1934-1501735-3.  Google Scholar

[1]

Amol Sasane. Extension of the $\nu$-metric for stabilizable plants over $H^\infty$. Mathematical Control & Related Fields, 2012, 2 (1) : 29-44. doi: 10.3934/mcrf.2012.2.29
[2]

Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157
[3]

Vitali Milman, Liran Rotem. $\alpha$-concave functions and a functional extension of mixed volumes. Electronic Research Announcements, 2013, 20: 1-11. doi: 10.3934/era.2013.20.1
[4]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019105
[5]

Daniel Alpay, Mihai Putinar, Victor Vinnikov. A Hilbert space approach to bounded analytic extension in the ball. Communications on Pure & Applied Analysis, 2003, 2 (2) : 139-145. doi: 10.3934/cpaa.2003.2.139
[6]

Chunrong Chen, Shengji Li. Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria. Journal of Industrial & Management Optimization, 2012, 8 (3) : 691-703. doi: 10.3934/jimo.2012.8.691
[7]

Ken Abe. Some uniqueness result of the Stokes flow in a half space in a space of bounded functions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 887-900. doi: 10.3934/dcdss.2014.7.887
[8]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[9]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[10]

Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353
[11]

Katrin Grunert, Helge Holden, Xavier Raynaud. Lipschitz metric for the Camassa--Holm equation on the line. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2809-2827. doi: 10.3934/dcds.2013.33.2809
[12]

Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041
[13]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[14]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167
[15]

Françoise Demengel, Thomas Dumas. Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian". Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1135-1155. doi: 10.3934/dcds.2019048
[16]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425
[17]

Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549
[18]

Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61
[19]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[20]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences