Whitney-type extensions in quasi-metric spaces

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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

Abstract
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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 and 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.
[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.
[3]	L. Chen, Smoothness and smooth extensions (I): Generalization of MWK functions and gradually varied functions, preprint, arXiv:1005.3727v1.
[4]	L. Chen, A digital-discrete method for smooth-continuous data reconstruction, preprint, arXiv:1010.3299v1.
[5]	R. R. Coifman and G. Weiss, "Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes," Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, 1971.
[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.
[7]	R. Engelking, "General Topology," Heldermann Verlag, Berlin, 1989.
[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.
[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.
[10]	H. Federer, "Geometric Measure Theory," Springer-Verlag, Berlin, 1969.
[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.
[12]	J. Heinonen, "Lectures on Analysis on Metric Spaces," Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.
[13]	E. Hille and R. S. Phillips, "Functional Analysis and Semigroups," Amer. Math. Soc. Colloq. Publ., Vol. 31, Amer. Math. Soc., Providence, RI, 1957.
[14]	L. Hörmander, On the division of distributions by polynomials, Ark. Mat., 3 (1958), 555-568. doi: 10.1007/BF02589517.
[15]	A. Jonsson and H. Wallin, "Function Spaces on Subsets of $R^n$," Math. Rep., Vol. 2, No. 1, 1984.
[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.
[17]	M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche transformationen, Fundamenta Mathematicae, 22 (1934), 77-108.
[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.
[19]	S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Expositiones Mathematicae, 3 (1983), 193-260.
[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.
[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.
[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.
[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.
[24]	J. R. Munkres, "Topology," Second edition, Prentice Hall, 2000.
[25]	S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 5 (1957), 471-473.
[26]	R. A. Rosenbaum, Subadditive functions, Duke Math. J., 17 (1950), 227-242. doi: 10.1215/S0012-7094-50-01721-2.
[27]	A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters, 4 (1986), 177-184. doi: 10.1016/0167-8655(86)90017-6.
[28]	J. T. Schwartz, "Nonlinear Functional Analysis," Gordon and Breach Science Publishers, New York, 1969.
[29]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970.
[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.
[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.
[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.

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References:

[1]	T. Aoki, Locally bounded linear topological spaces, Proc. Japan Acad. Tokyo, 18 (1942), 588-594. doi: 10.3792/pia/1195573733.
[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.
[3]	L. Chen, Smoothness and smooth extensions (I): Generalization of MWK functions and gradually varied functions, preprint, arXiv:1005.3727v1.
[4]	L. Chen, A digital-discrete method for smooth-continuous data reconstruction, preprint, arXiv:1010.3299v1.
[5]	R. R. Coifman and G. Weiss, "Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes," Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, 1971.
[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.
[7]	R. Engelking, "General Topology," Heldermann Verlag, Berlin, 1989.
[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.
[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.
[10]	H. Federer, "Geometric Measure Theory," Springer-Verlag, Berlin, 1969.
[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.
[12]	J. Heinonen, "Lectures on Analysis on Metric Spaces," Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.
[13]	E. Hille and R. S. Phillips, "Functional Analysis and Semigroups," Amer. Math. Soc. Colloq. Publ., Vol. 31, Amer. Math. Soc., Providence, RI, 1957.
[14]	L. Hörmander, On the division of distributions by polynomials, Ark. Mat., 3 (1958), 555-568. doi: 10.1007/BF02589517.
[15]	A. Jonsson and H. Wallin, "Function Spaces on Subsets of $R^n$," Math. Rep., Vol. 2, No. 1, 1984.
[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.
[17]	M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche transformationen, Fundamenta Mathematicae, 22 (1934), 77-108.
[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.
[19]	S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Expositiones Mathematicae, 3 (1983), 193-260.
[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.
[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.
[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.
[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.
[24]	J. R. Munkres, "Topology," Second edition, Prentice Hall, 2000.
[25]	S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 5 (1957), 471-473.
[26]	R. A. Rosenbaum, Subadditive functions, Duke Math. J., 17 (1950), 227-242. doi: 10.1215/S0012-7094-50-01721-2.
[27]	A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters, 4 (1986), 177-184. doi: 10.1016/0167-8655(86)90017-6.
[28]	J. T. Schwartz, "Nonlinear Functional Analysis," Gordon and Breach Science Publishers, New York, 1969.
[29]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970.
[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.
[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.
[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.

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