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Complex interpolation and Calderón-Mityagin couples of Morrey spaces. (English) Zbl 1435.46018

Summary: We study interpolation spaces between global Morrey spaces and between local Morrey spaces. We prove that for a wide class of couples of these spaces the upper complex Calderón spaces are not described by the \(K\)-method of interpolation. A by-product of our results is that couples of Morrey spaces belonging to this class are not Calderón-Mityagin couples. A Banach couple \((X_0,X_1)\) is said to have the universal \(K\)-property if all relative interpolation spaces from any Banach couple to \((X_0,X_1)\) are relatively \(K\)-monotone. A couple of local Morrey spaces is proved to have the universal \(K\)-property once it is a Calderón-Mityagin couple.

MSC:

46B70 Interpolation between normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B35 Function spaces arising in harmonic analysis
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[1] 10.1512/iumj.2004.53.2470 · Zbl 1100.31009 · doi:10.1512/iumj.2004.53.2470
[2] 10.1007/s11512-010-0134-0 · Zbl 1254.31009 · doi:10.1007/s11512-010-0134-0
[3] 10.1512/iumj.1979.28.28054 · Zbl 0394.41004 · doi:10.1512/iumj.1979.28.28054
[4] 10.1007/978-3-642-66451-9 · doi:10.1007/978-3-642-66451-9
[5] ; Beurling, Ann. Inst. Fourier (Grenoble), 14, 1 (1964) · Zbl 0133.07501
[6] ; Brudnyi, Interpolation functors and interpolation spaces, I. North-Holland Math. Library, 47 (1991) · Zbl 0743.46082
[7] ; Brudnyĭ, Dokl. Akad. Nauk SSSR, 256, 14 (1981)
[8] 10.4064/sm163-2-4 · Zbl 1044.42015 · doi:10.4064/sm163-2-4
[9] 10.4064/sm-24-2-113-190 · Zbl 0204.13703 · doi:10.4064/sm-24-2-113-190
[10] 10.4064/sm-26-3-301-304 · doi:10.4064/sm-26-3-301-304
[11] ; Campanato, Ann. Scuola Norm. Sup. Pisa (3), 19, 87 (1965)
[12] 10.1007/BF02384473 · Zbl 0467.46061 · doi:10.1007/BF02384473
[13] 10.1007/BF02384370 · Zbl 0535.46049 · doi:10.1007/BF02384370
[14] ; Cwikel, Studia Math., 138, 209 (2000)
[15] 10.1090/memo/0787 · Zbl 1044.46018 · doi:10.1090/memo/0787
[16] ; Cwikel, J. Math. Pures Appl. (9), 60, 1 (1981) · Zbl 0415.46052
[17] 10.1017/CBO9780511897207 · doi:10.1017/CBO9780511897207
[18] 10.1112/plms/s3-69.3.605 · Zbl 0831.42012 · doi:10.1112/plms/s3-69.3.605
[19] 10.4064/sm-106-3-233-277 · Zbl 0810.46020 · doi:10.4064/sm-106-3-233-277
[20] 10.1016/C2013-0-03044-7 · doi:10.1016/C2013-0-03044-7
[21] ; Kreĭn, Interpolation of linear operators. Trans. Math. Monographs, 54 (1982) · Zbl 0493.46058
[22] ; Lemarié-Rieusset, Eurasian Math. J., 3, 62 (2012) · Zbl 1277.46017
[23] 10.1007/s11118-012-9295-8 · Zbl 1267.42024 · doi:10.1007/s11118-012-9295-8
[24] ; Lozanovskii, Sibirsk. Mat. Ž., 10, 584 (1969) · Zbl 0184.34801
[25] ; Lozanovskii, Funkcional. Anal. i Priložen., 6, 89 (1972)
[26] ; Lozanovskii, Qualitative and approximate methods for the investigation of operator equations, III, 122 (1978)
[27] 10.4064/sm-125-3-201-218 · Zbl 0910.46058 · doi:10.4064/sm-125-3-201-218
[28] 10.1016/j.jfa.2017.01.016 · Zbl 1457.46025 · doi:10.1016/j.jfa.2017.01.016
[29] 10.1090/S0002-9947-02-03214-2 · Zbl 1022.35039 · doi:10.1090/S0002-9947-02-03214-2
[30] ; Mityagin, Mat. Sb., 66(108), 473 (1965) · Zbl 0142.10701
[31] 10.2307/1989904 · Zbl 0018.40501 · doi:10.2307/1989904
[32] 10.4064/sm188-3-1 · Zbl 1163.46020 · doi:10.4064/sm188-3-1
[33] ; Ovchinnikov, Math. Rep., 1, 349 (1984) · Zbl 0875.46007
[34] 10.1016/0022-1236(69)90022-6 · Zbl 0175.42602 · doi:10.1016/0022-1236(69)90022-6
[35] ; Reisner, Bull. Polish Acad. Sci. Math., 41, 189 (1993) · Zbl 0809.46019
[36] ; Sedaev, Optimizacija, 4(21), 98 (1971)
[37] ; Shestakov, Vestnik Leningrad Univ., 19, 64 (1974) · Zbl 0297.46028
[38] 10.1002/cpa.3160170303 · Zbl 0149.09201 · doi:10.1002/cpa.3160170303
[39] 10.1007/978-1-4612-1972-9_6 · doi:10.1007/978-1-4612-1972-9_6
[40] 10.1007/BF02546511 · JFM 56.0954.02 · doi:10.1007/BF02546511
[41] 10.2307/1968102 · Zbl 0004.05905 · doi:10.2307/1968102
[42] 10.1007/s11425-015-5047-8 · Zbl 1337.46030 · doi:10.1007/s11425-015-5047-8
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