Abstract
The paper deals with interpolation orbits for linear operators acting from a arbitrary couple {\(L_{p_0 }\) (U 0), \(L_{p_1 }\) (U 1)} of weighted L p spaces into an arbitrary couple {\(L_{q_0 }\) (V 0), \(L_{q_1 }\) (V 1)} of such spaces, where 1 ⩽ p 0,p 1,q 0,q 1 ⩽ ∞. Here L p (U) is the space of measurable functions f on a measure space such that fU ∈ L p , equipped with the norm \(\parallel {\text{f}}\parallel _{Lp(U)} \; = \;\parallel {\text{f}}\parallel _{Lp}\). The paper describes the orbits of arbitrary elements a ∈ \(L_{p_0 }\) (U 0) + \(L_{p_1 }\) (U 1). It contains proofs of the results announced in C. R. Acad. Sci. Paris, Ser. I, 334, 881–884 (2002).
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References
G. Bennett, “Inclusion mappings between l p spaces,” J. Funct. Anal., 13, 20–27 (1973).
Ju. A. Brudnyi and N. Ya. Krugliak, Interpolation Functors and Interpolation Spaces. I, North Holland, Amsterdam, 1991.
Ju. A. Brudnyi and A. Shteinberg, “Calderon couples of Lipschitz spaces,” J. Funct. Anal., 131, 459–498 (1995).
V. I. Dmitriev, “On interpolation of operators in L p spaces,” Dokl. Akad. Nauk SSSR, 260, 1051–1054 (1981); English transl. Soviet Math. Dokl., 24, 373–376 (1981).
S. Janson, “Minimal and maximal methods in interpolation,” J. Funct. Anal., 44, 50–73 (1981).
J.-L. Lions and J. Peetre, “Sur une classe d’espaces d’interpolation,” Inst. Hautes Études Sci. Publ. Math., 19, 5–68 (1964).
G. Ja. Lozanovskii, “On some Banach structures, II,” Sib. Mat. Zh., 12, 562–567 (1971).
V. I. Ovchinnikov, “An exact interpolation theorem in L p spaces,” Dokl. Akad. Nauk SSSR, 272, 300–303 (1983); English transl. Soviet Math. Dokl., 28, 381–385 (1983).
V. I. Ovchinnikov, “Description of orbits,” In: Israel Math. Conf. Proc., Vol. 5, 1992, pp. 291–292.
V. I. Ovchinnikov, “On the description of interpolation orbits i couples of L p spaces, when they are ot described by the K-method,” In: Israel Math. Conf. Proc., Vol. 5, 1992, pp. 187–206.
V. I. Ovchinnikov, “The method of orbits in interpolation theory,” Math. Rep., 1, part 2, 349–516 (1984).
V. I. Ovchinnikov, “Interpolation orbits in couples of L p spaces,” C. R. Acad. Sci. Paris, Ser. I, 334, 881–884 (2002).
G. Sparr, “Interpolation des espaces L p w ,” C. R. Acad. Sci. Paris, Ser. A–B, 278, 491–492 (1974).
G. Sparr, “Interpolation of weighted L p spaces,” Studia Math. 62, 229–271 (1978).
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 56–68, 2005
Original Russian Text Copyright © by V. I. Ovchinnikov
Translated by V. I. Ovchinnikov
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Ovchinnikov, V.I. Interpolation orbits in couples of Lebesgue spaces. Funct Anal Its Appl 39, 46–56 (2005). https://doi.org/10.1007/s10688-005-0016-6
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DOI: https://doi.org/10.1007/s10688-005-0016-6