Abstract
We obtain necessary and sufficient conditions for a homeomorphism of domains in a Euclidean space to generate a bounded embedding operator of the Orlicz–Sobolev spaces defined by a special class of N-functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Vodop’yanov S. K., The Taylor Formula and Function Spaces [Russian], Novosibirsk Univ., Novosibirsk (1988).
Vodop’yanov S. K., “Mappings of homogeneous groups and imbeddings of functional spaces,” Sib. Math. J., 30, No. 5, 685–698 (1989).
Ukhlov A. D., “On mappings generating the embeddings of Sobolev spaces,” Sib. Math. J., 34, No. 1, 165–171 (1993).
Vodop’yanov S. K. and Ukhlov A. D., “Sobolev spaces and (P,Q)-quasiconformal mappings of Carnot groups,” Sib. Math. J., 39, No. 4, 665–682 (1998).
Vodop’yanov S. K. and Ukhlov A. D., “Superposition operators in Sobolev spaces,” Russian Math. (Iz. VUZ), 46, No. 10, 9–31 (2002).
Vodop’yanov S. K., “Regularity of mappings inverse to Sobolev mappings,” Sb. Math., 203, No. 10, 1383–1410 (2012).
Sobolev S. L., “On some transformation groups of an n-dimensional space,” Dokl. Akad. Nauk SSSR, 32, No. 6, 380–382 (1941).
Maz’ya V. G., Classes of Sets and Embedding Theorems of Function Classes [Russian], Diss. Kand. Fiz.-Mat. Nauk., LGU, Leningrad (1961).
Gehring F. W., “Lipschitz mappings and the p-capacity of rings in n-space,” in: Advances in the Theory of Riemann Surfaces (Proc. Conf. Stony Brook, New York, 1969), Princeton Univ. Press, Princeton, 1971, 66, pp. 175–193.
Vodop’yanov S. K. and Gol’dshteĭn V. M., “Lattice isomorphisms of the spaces W1 n and quasiconformal mappings,” Sib. Math. J., 16, No. 2, 174–189 (1975).
Hencl S. and Kleprlik L., “Composition of q-quasiconformal mappings and functions in Orlicz–Sobolev spaces,” Illinois J. Math., 56, No. 3, 931–955 (2012).
Krasnosel’skiĭ M. A. and Rutitskiĭ Ya. B., Convex Functions and Orlicz Spaces, P. Noordhoff Ltd, Groningen (1961).
Chernov A. V., “An analog of the generalized Hölder inequality in Orlicz spaces,” Vestn. Nizhegor. Univ., 6, No. 1, 157–161 (2013).
Rao M. M. and Ren Z. D., Theory of Orlicz Spaces, Marcel Dekker, New York (1991) (Pure Appl. Math.).
Donaldson T. K. and Trudinger N. S., “Orlicz–Sobolev spaces and imbedding theorems,” J. Funct. Anal., 8, No. 1, 52–75 (1971).
Hajlasz P., “Change of variables formula under minimal assumptions,” Colloq. Math., 64, No. 1, 93–101 (1993).
Maly J., Swanson D., and Ziemer W. P., “Fine behavior of functions whose gradients are in an Orlicz space,” Stud. Math., 190, No. 1, 33–71 (2009).
Vodopyanov S. K. and Ukhlov A. D., “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. II,” Mat. Tr., 7, No. 1, 13–49 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was partially supported by the Government of the Russian Federation (Agreement 14.B25.31.0029) and the Russian Foundation for Basic Research (Grant 14–01–00552).
Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 5, pp. 1088–1101, September–October, 2016; DOI: 10.17377/smzh.2016.57.514.
Rights and permissions
About this article
Cite this article
Menovshchikov, A.V. Composition operators in Orlicz–Sobolev spaces. Sib Math J 57, 849–859 (2016). https://doi.org/10.1134/S0037446616050141
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446616050141