Abstract
We study the boundary values of the functions of the Sobolev function spaces W l∞ and the Nikol’skiĭ function spaces H l∞ which are defined on an arbitrary domain of a Carnot group. We obtain some reversible characteristics of the traces of the spaces under consideration on the boundary of the domain of definition and sufficient conditions for extension of the functions of these spaces outside the domain of definition. In some cases these sufficient conditions are necessary.
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References
G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, vol. 28 of Math. Notes (Princeton University Press, Princeton, N.J., 1982).
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 (Princeton University Press, Princeton, N.J., 1970).
S. K. Vodop’yanov, “Intrinsic geometries and spaces of differentiable functions,” in Functional Analysis and Mathematical Physics (Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1987), pp. 18–38.
S. K. Vodop’yanov, “Isoperimetric relations and conditions for the extension of differentiable functions,” Dokl. Akad. Nauk SSSR 292(1), 11–15 (1987) [Soviet Math. Dokl. 35, 1–5 (1987)].
S. K. Vodop’yanov, “Equivalent normalizations of Sobolev and Nikol’skiĭ spaces in domains. Boundary values and extension,” in Function Spaces and Applications (Lund, 1986) (Springer, Berlin, 1988), vol. 1302 of Lecture Notes in Math., pp. 397–409.
S. K. Vodop’yanov, Taylor’s Formula and Function Spaces (Novosibirsk State Univ., Novosibirsk, 1988).
S. K. Vodop’yanov, “Boundary behavior of differentiable functions and related topics,” in Nonlinear Analysis, Function Spaces and Applications (Roudnice nad Labem, 1990) (Teubner, Leipzig, 1990), vol. 4 of Teubner-Texte Math., Bd 119, pp. 224–253.
S. K. Vodop’yanov and A. V. Greshnov, “On the continuation of functions of bounded mean oscillation on spaces of homogeneous type with intrinsic metric,” Sibirsk. Mat. Zh. 36(5), 1015–1048 (1995) [Siberian Math. J. 36 (5), 873–902 (1995)].
S. K. Vodop’yanov and I. M. Pupyshev, “Whitney-Type theorems of extension of functions on Carnot groups,” Sibirsk. Mat. Zh. 47(4), 731–752 (2006) [Siberian Math. J. 47 (4), 601–620 (2006)].
H. Whitney, “Analytic extensions of differentiable functions defined in closed sets,” Trans. Amer. Math. Soc. 36(1), 63–89 (1934).
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Dedicated to Yu. G. Reshetnyak on the occasion of his 7th birthday
Original Russian Text © S. K. Vodop’yanov and I. M. Pupyshev, 2006, published in Matematicheskie Trudy, 2006, Vol. 9, No. 2, pp. 23–46.
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Vodop’yanov, S.K., Pupyshev, I.M. Boundary values of differentiable functions defined on an arbitrary domain of a Carnot group. Sib. Adv. Math. 17, 62–78 (2007). https://doi.org/10.3103/S105513440701004X
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DOI: https://doi.org/10.3103/S105513440701004X