Arkiv för Matematik

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  • Volume 40, Number 2 (2002), 363-382.

Sobolev spaces in several variables in L1-type norms are not isomorphic to Banach lattices

Aleksander Pełczyński and Michał Wojciechowski

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A Sobolev space in several variables in an L1-type norm is not complemented in its second dual. Hence it is not isomorphic as a Banach space to any complemented subspace of a Banach lattice.


Both authors were supported in part by the Polish KBN grant 2P03A 036 14.

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Ark. Mat., Volume 40, Number 2 (2002), 363-382.

Received: 16 May 2001
First available in Project Euclid: 31 January 2017

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2002 © Institut Mittag-Leffler


Pełczyński, Aleksander; Wojciechowski, Michał. Sobolev spaces in several variables in L 1 -type norms are not isomorphic to Banach lattices. Ark. Mat. 40 (2002), no. 2, 363--382. doi:10.1007/BF02384541.

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  • [A] Adams, R., Sobolev Spaces, Academic Press, New York, 1975.
  • [BIN] Besov, O. V., Il’in, V. P. and Nikolskiî, S. M., Integral Representations of Functions and Embedding Theorems, 2nd ed., Nauka, Moscow, 1996, (Russian) English transl. of 1st ed.: vol. 1, Winston, Washington, D. C., 1978.
  • [B] Borsuk, K., Uber Isomorphie der Funktionalräume, Bull. Int. Acad. Polon. Sci. Sl. Sci. Math. Nat. Sér. A 1933 (1933), 1–10.
  • [BS] Brudnyi, Y. and Shvartsman, P., The Whitney Extension Problem, Manuscript, Haifa, 1999.
  • [DJP] Diestel, J., Jarchow, H. and Tonge, A., Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995.
  • [D] Dixmier, J., Sur une théorème de Banach, Duke Math. J. 15 (1948), 1057–1071.
  • [DS] Dunford, N. and Schwartz, J. T., Linear Operators, Part I, Interscience, New York, 1958.
  • [G] Gagliardo, E., Caratterizazioni delle trace sulla frontiera relative ad alcune classi funzioni in piu variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284–305.
  • [Gr] Grothendieck, A., Erratum au mémoire: Produits tensoriels topologiques et espaces nucléaires, Ann. Inst. Fourier (Grenoble) 6 (1956), 117–120.
  • [H] Henkin, G. M., The nonisomorphy of certain spaces of functions of different number of variables, Funktsional. Anal. i Prilozhen. 1:4 (1967), 57–78 (Russian). English transl.: Funct. Anal. Appl. 1 (1967), 306–315.
  • [KaP] Kalton, N. and Pełczyński, A., Kernels of surjections from L1-spaces with an applications to the Sidon sets, Math. Ann. 309 (1997), 135–158.
  • [K1] Kislyakov, S. V., Sobolev embedding operators and the nonisomorphism of certain Banach spaces, Funktsional. Anal. i Prilozhen. 9:4 (1975), 22–27 (Russian). English transl.: Funct. Anal. Appl. 9 (1975), 290–294.
  • [K2] Kislyakov, S. V., In the space of continuously differentiable function on the torus there is no local unconditional structure, Preprint LOMI P1-77, 1977.
  • [Ko] Kolyada, V., Rearrangements of functions and embedding of anisotropic spaces of Sobolev type, East J. Approx. 4 (1998), 111–199, 431.
  • [KwP] Kwapień, S. and Pełczyński, A., Absolutely summing operators and translation-invariant spaces of functions on compact abelian groups, Math. Nachr. 94 (1980), 303–340.
  • [L] Lindenstrauss, J., On a certain subspace of l1, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12 (1964), 539–542.
  • [L] Lindenstrauss, J. and Tzafriri, J., Classical Banach Spaces I, II, Springer-Verlag, Berlin-Heidelberg, 1977, 1979.
  • [M] Maz’ya, V. G., Sobolev Spaces, Springer-Verlag, Berlin, 1985.
  • [P] Peetre, J., A counterexample connected with Gagliardo’s trace theorem, Comment. Math., Special Issue 2 (1979), 277–282.
  • [PS] Pełczyński, A. and Senator, K., On isomorphisms of anisotropic Sobolev spaces with “classical Banach spaces” and a Sobolev type embedding theorem, Studia Math. 84 (1986), 169–215.
  • [PW1] Pełczyński, A. and Wojciechowski, M., Contribution to the isomorphic classification of Sobolev spaces L(k)p (Ω) (1≤p<∞), in Recent Progress in Functional Analysis, Proceedings Valdivia Conference (Valencia, 2000) (Bierstedt, K. D., Bonet, J., Maestre, J. and Schmets, J., eds.), North-Holland Math. Studies 189, North-Holland, Amsterdam, 2001.
  • [PW2] Pełczyński, A. and Wojciechowski, M., Sobolev spaces, in Handbook of the Geometry of Banach Spaces, Vol. II (Johnson, W. B. and Lindenstrauss, J., eds.), North Holland, Amsterdam, to appear.
  • [S] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.
  • [SW] Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.