The quasinormed Neumann–Schatten ideals and embedding theorems for the generalized Lions–Peetre spaces of means
HTML articles powered by AMS MathViewer
- by
V. I. Ovchinnikov
Translated by: the author - St. Petersburg Math. J. 22 (2011), 669-681
- DOI: https://doi.org/10.1090/S1061-0022-2011-01162-8
- Published electronically: May 3, 2011
- PDF | Request permission
Abstract:
For the spaces $\varphi (X_0,X_1)_{p_0,p_1}$, which generalize the spaces of means introduced by Lions and Peetre to the case of functional parameters, necessary and sufficient conditions are found for embedding when all parameters (the function $\varphi$ and the numbers $1\leq p_0$, $p_1\leq \infty )$ vary. The proof involves a description of generalized Lions–Peetre spaces in terms of orbits and co-orbits of von Neumann–Schatten ideals (including quasinormed ideals).References
- N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation methods, Ann. Mat. Pura Appl. (4) 68 (1965), 51–117. MR 226361, DOI 10.1007/BF02411022
- Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation functors and interpolation spaces. Vol. I, North-Holland Mathematical Library, vol. 47, North-Holland Publishing Co., Amsterdam, 1991. Translated from the Russian by Natalie Wadhwa; With a preface by Jaak Peetre. MR 1107298
- Yuri Brudnyi and Alexander Shteinberg, Calderón couples of Lipschitz spaces, J. Funct. Anal. 131 (1995), no. 2, 459–498. MR 1345039, DOI 10.1006/jfan.1995.1096
- V. A. Dikarev and V. I. Macaev, An exact interpolation theorem, Dokl. Akad. Nauk SSSR 168 (1966), 986–988 (Russian). MR 0201963
- A. A. Dmitriev and E. M. Semënov, Optimality of the M. Riesz interpolation theorem in the upper triangle, Dokl. Akad. Nauk SSSR 258 (1981), no. 6, 1298–1300 (Russian). MR 625747
- Svante Janson, Minimal and maximal methods of interpolation, J. Functional Analysis 44 (1981), no. 1, 50–73. MR 638294, DOI 10.1016/0022-1236(81)90004-5
- E. D. Kravishvili, Method of means with arbitrary functional parameter, Trudy Mat. Fak. Voronezh. Gos. Univ. (N.S.) No. 7 (2002), 58–72. (Russian)
- E. D. Kravishvili and V. I. Ovchinnikov, Description of the interpolation orbits of the Neumann-Schatten ideals acting in Hilbert couples, and embedding theorems, Dokl. Akad. Nauk 393 (2003), no. 1, 10–13 (Russian). MR 2106705
- J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343
- Mario Milman, The computation of the $K$ functional for couples of rearrangement invariant spaces, Results Math. 5 (1982), no. 2, 174–176. MR 685875, DOI 10.1007/BF03323313
- V. I. Ovchinnikov, Exact interpolation theorem for $L_{p}$ spaces, Dokl. Akad. Nauk SSSR 272 (1983), no. 2, 300–303 (Russian). MR 724507
- V. I. Ovchinnikov, Interpolation orbits in couples of Lebesgue spaces, Funktsional. Anal. i Prilozhen. 39 (2005), no. 1, 56–68, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 39 (2005), no. 1, 46–56. MR 2132439, DOI 10.1007/s10688-005-0016-6
- V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Rep. 1 (1984), no. 2, i–x and 349–515. MR 877877
- Vladimir I. Ovchinnikov, Interpolation orbits in couples of $L_p$ spaces, C. R. Math. Acad. Sci. Paris 334 (2002), no. 10, 881–884 (English, with English and French summaries). MR 1909932, DOI 10.1016/S1631-073X(02)02351-8
- Vladimir I. Ovchinnikov, Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters, J. Funct. Anal. 228 (2005), no. 1, 234–243. MR 2171790, DOI 10.1016/j.jfa.2005.03.010
- A. A. Sedaev, A description of the interpolation spaces of the couple $(L^{p}_{a_{0}},\,L^{p}_{a_{1}})$, and certain related questions, Dokl. Akad. Nauk SSSR 209 (1973), 798–800 (Russian). MR 0324396
- M. Riesz, Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires, Acta Math. 49 (1926), 465–497.
Bibliographic Information
- V. I. Ovchinnikov
- Affiliation: Voronezh State University, Universitetskaya Ploshchad’, 1, Voronezh 394006, Russia
- Email: vio@comch.ru
- Received by editor(s): May 20, 2009
- Published electronically: May 3, 2011
- Additional Notes: Supported by RFBR (grant no. 07-01-00131)
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 669-681
- MSC (2010): Primary 46M35, 46B70
- DOI: https://doi.org/10.1090/S1061-0022-2011-01162-8
- MathSciNet review: 2768965