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1993 | 104 | 2 | 161-180
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A Carlson type inequality with blocks and interpolation

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An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of "blocks" and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function (see [6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre's interpolation functor $⟨⟩_{φ}$ (see [26]) and its Gagliardo closure on couples of functional Banach lattices in terms of the Calderón-Lozanovskiǐ construction. Our interest in this functor is inspired by the fact that if $φ = t^{θ}(0 < θ < 1)$, then, on couples of Banach lattices and their retracts, it coincides with the complex method (see [20], [27]) and, thus, it may be regarded as a "real version" of the complex method.
Twórcy
  • Department of Mathematics, Yaroslavl State University, Sovetskaya 14, 150 000 Yaroslavl, Russia
  • Department of Mathematics, Luleå University, S-951 87 Luleå, Sweden
  • Department of Mathematics, Luleå University, S-951 87 Luleå, Sweden
Bibliografia
  • [1] N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation methods, Ann. Mat. Pura Appl. 68 (1965), 51-117.
  • [2] E. F. Beckenbach and R. Bellman, Inequalities, Springer, 1983.
  • [3] E. I. Berezhnoǐ, Interpolation of linear and compact operators in $φ(X_0,X_1)$ spaces, in: Qualitative and Approximative Methods for the Investigation of Operator Equations 5, Yaroslavl 1980, 19-29 (in Russian).
  • [4] J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976.
  • [5] Yu. A. Brudnyǐ and N. Ya. Kruglyak, Interpolation Functors, book manuscript, Yaroslavl 1981 (in Russian).
  • [6] Yu. A. Brudnyǐ and N. Ya. Kruglyak, Interpolation Functors and Interpolation Spaces I, North-Holland, 1991.
  • [7] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 133-190.
  • [8] F. Carlson, Une inégalité, Ark. Mat. Astr. Fysik 25 B (1934), 1-5.
  • [9] E. Gagliardo, Caratterizzazione costruttiva di tutti gli spazi interpolazione tra spazi di Banach, in: Symposia Math. 2 (INDAM, Rome 1968), Academic Press, London 1969, 95-106.
  • [10] J. Gustavsson, On interpolation of weighted $L^p$-spaces and Ovchinnikov's theorem, Studia Math. 72 (1982), 237-251.
  • [11] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, ibid. 60 (1977), 33-59.
  • [12] G. H. Hardy, A note on two inequalities, J. London Math. Soc. 11 (1936), 167-170.
  • [13] S. Janson, Minimal and maximal methods of interpolation, J. Funct. Anal. 44 (1981), 50-73.
  • [14] B. Kjellberg, On some inequalities, in: C. R. Dixième Congrès des Mathématiciens Scandinaves 1946, Jul. Gjellerups Forlag, Copenhagen 1946, 333-340.
  • [15] N. Ya. Kruglyak, A new proof of the Riesz-Thorin theorem and interpolation property of the Calderón-Lozanovskiǐ construction, research report, Yaroslavl 1983 (VINITI No. 6909-83 Dep.) (in Russian).
  • [16] N. Ya. Kruglyak and M. Mastyło, Correct interpolation functors of orbits, J. Funct. Anal., to appear.
  • [17] G. Ya. Lozanovskiǐ, A remark on an interpolation theorem of Calderón, Funktsional. Anal. i Prilozhen. 6 (4) (1972), 89-90; English transl.: Functional Anal. Appl. 6 (1972), 333-334.
  • [18] G. Ya. Lozanovskiǐ, On some Banach lattices. IV, Sibirsk. Mat. Zh. 14 (1973), 140-155 (in Russian).
  • [19] G. Ya. Lozanovskiǐ, On a complex method of interpolation in Banach lattices of measurable functions, Dokl. Akad. Nauk SSSR 226 (1976), 55-57; English transl.: Soviet Math. Dokl. 17 (1) (1976), 51-54.
  • [20] G. Ya. Lozanovskiǐ, The complex method of interpolation in Banach lattices of measurable functions, in: Probl. Mat. Anal. 7, Leningrad 1979, 83-99 (in Russian)
  • [21] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Math. 5, Campinas 1989.
  • [22] P. Nilsson, Interpolation of Banach lattices, Studia Math. 82 (1985), 133-154.
  • [23] V. I. Ovchinnikov, Interpolation theorems following from Grothendieck's inequality, Funktsional. Anal. i Prilozhen. 10 (4) (1976), 45-54; English transl.: Functional Anal. Appl. 10 (1976), 287-294 (1977).
  • [24] V. I. Ovchinnikov, Interpolation of quasi-normed Orlicz spaces, Funktsional. Anal. i Prilozhen. 16 (3) (1982), 78-79; English transl.: Functional Anal. Appl. 16 (1982), 223-224 (1983).
  • [25] V. I. Ovchinnikov, The Method of Orbits in Interpolation Theory, Math. Rep. 1, Part 2, Harwood, 1984.
  • [26] J. Peetre, Sur l'utilisation des suites inconditionnellement sommables dans la théorie des espaces d'interpolation, Rend. Sem. Mat. Univ. Padova 46 (1971), 173-190.
  • [27] V. A. Shestakov, Complex interpolation in Banach spaces of measurable functions, Vestnik Leningrad. Univ. 1974 (19), 64-68 (in Russian).
  • [28] V. A. Shestakov, Transformations of Banach ideal spaces and the interpolation of linear operators, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), 569-577 (in Russian).
  • [29] P. P. Zabreǐko, An interpolation theorem for linear operators, Mat. Zametki 2 (1967), 593-598; English transl. in Math. Notes 2 (1967).
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Bibliografia
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