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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), 588-594. 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), 2487-2512. 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, Springer-Verlag, 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), 569-645. doi: 10.1090/S0002-9904-1977-14325-5.  Google Scholar

[7]

R. Engelking, "General Topology," Heldermann Verlag, Berlin, 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), 423-446. doi: 10.4171/RMI/571.  Google Scholar

[9]

C. Fefferman, A sharp form of Whitney's extension theorem, Ann. of Math., 161 (2005), 509-577. doi: 10.4007/annals.2005.161.509.  Google Scholar

[10]

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

[11]

A. Gogatishvili, P. Koskela and N. Shanmugalingam, Interpolation properties of Besov spaces defined on metric spaces, Mathematische Nachrichten, Special Issue: Erhard Schmidt Memorial Issue, Part II, Vol. 283 (2010), 215-231. doi: 10.1002/mana.200810242.  Google Scholar

[12]

J. Heinonen, "Lectures on Analysis on Metric Spaces," Springer-Verlag, New York, 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, Amer. Math. Soc., Providence, RI, 1957.  Google Scholar

[14]

L. Hörmander, On the division of distributions by polynomials, Ark. Mat., 3 (1958), 555-568. doi: 10.1007/BF02589517.  Google Scholar

[15]

A. Jonsson and H. Wallin, "Function Spaces on Subsets of $R^n$," Math. Rep., Vol. 2, No. 1, 1984. Google Scholar

[16]

N. J. Kalton, N. T. Peck and J. W. Roberts, "An $F$-space Sampler," London Math. Society Lecture Notes Series, Vol. 89, Cambridge University Press, Cambridge, 1984. Google Scholar

[17]

M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche transformationen, Fundamenta Mathematicae, 22 (1934), 77-108. Google Scholar

[18]

P. Koskela, N. Shanmugalingam and H. Tuominen, Removable sets for the Poincaré inequality on metric spaces, Indiana Math. J., 49 (2000), 333-352. 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), 193-260.  Google Scholar

[20]

R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math., 33 (1979), 257-270. 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), 271-309. 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), 837-842. 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, Brikhäuser (2012), 481 pages. Google Scholar

[24]

J. R. Munkres, "Topology," Second edition, Prentice Hall, 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), 471-473.  Google Scholar

[26]

R. A. Rosenbaum, Subadditive functions, Duke Math. J., 17 (1950), 227-242. doi: 10.1215/S0012-7094-50-01721-2.  Google Scholar

[27]

A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters, 4 (1986), 177-184. doi: 10.1016/0167-8655(86)90017-6.  Google Scholar

[28]

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

[29]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 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), 100-108. 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), 83-93. doi: 10.2307/2371917.  Google Scholar

[32]

H. Whitney, Analytic extensions of functions defined on closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. 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), 588-594. 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), 2487-2512. 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, Springer-Verlag, 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), 569-645. doi: 10.1090/S0002-9904-1977-14325-5.  Google Scholar

[7]

R. Engelking, "General Topology," Heldermann Verlag, Berlin, 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), 423-446. doi: 10.4171/RMI/571.  Google Scholar

[9]

C. Fefferman, A sharp form of Whitney's extension theorem, Ann. of Math., 161 (2005), 509-577. doi: 10.4007/annals.2005.161.509.  Google Scholar

[10]

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

[11]

A. Gogatishvili, P. Koskela and N. Shanmugalingam, Interpolation properties of Besov spaces defined on metric spaces, Mathematische Nachrichten, Special Issue: Erhard Schmidt Memorial Issue, Part II, Vol. 283 (2010), 215-231. doi: 10.1002/mana.200810242.  Google Scholar

[12]

J. Heinonen, "Lectures on Analysis on Metric Spaces," Springer-Verlag, New York, 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, Amer. Math. Soc., Providence, RI, 1957.  Google Scholar

[14]

L. Hörmander, On the division of distributions by polynomials, Ark. Mat., 3 (1958), 555-568. doi: 10.1007/BF02589517.  Google Scholar

[15]

A. Jonsson and H. Wallin, "Function Spaces on Subsets of $R^n$," Math. Rep., Vol. 2, No. 1, 1984. Google Scholar

[16]

N. J. Kalton, N. T. Peck and J. W. Roberts, "An $F$-space Sampler," London Math. Society Lecture Notes Series, Vol. 89, Cambridge University Press, Cambridge, 1984. Google Scholar

[17]

M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche transformationen, Fundamenta Mathematicae, 22 (1934), 77-108. Google Scholar

[18]

P. Koskela, N. Shanmugalingam and H. Tuominen, Removable sets for the Poincaré inequality on metric spaces, Indiana Math. J., 49 (2000), 333-352. 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), 193-260.  Google Scholar

[20]

R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math., 33 (1979), 257-270. 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), 271-309. 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), 837-842. 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, Brikhäuser (2012), 481 pages. Google Scholar

[24]

J. R. Munkres, "Topology," Second edition, Prentice Hall, 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), 471-473.  Google Scholar

[26]

R. A. Rosenbaum, Subadditive functions, Duke Math. J., 17 (1950), 227-242. doi: 10.1215/S0012-7094-50-01721-2.  Google Scholar

[27]

A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters, 4 (1986), 177-184. doi: 10.1016/0167-8655(86)90017-6.  Google Scholar

[28]

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

[29]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 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), 100-108. 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), 83-93. doi: 10.2307/2371917.  Google Scholar

[32]

H. Whitney, Analytic extensions of functions defined on closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. doi: 10.1090/S0002-9947-1934-1501735-3.  Google Scholar

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