Geodesic
About estimates of the $\mathscr{K}$-divisibility constant for Banach pairs
Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 2, pp. 179-191.

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The paper contains some results on estimates of the $\mathscr{K}$-divisibility constant for Banach pairs. Its has been established that it is impossible to improve the estimate $3+2\sqrt2$ for any Banach pair and $4$ any pair of Banach lattices using the method of Yu. A. Brudnyi and N. Ya. Krugljak. I give a proof of Sedaev–Semenov theorem for the pair $(L_1^1,L_1)$ with measure on half-axis, using only the properties of concave functions. 
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A. A. Dmitriev. About estimates of the $\mathscr{K}$-divisibility constant for Banach pairs. Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 2, pp. 179-191. http://geodesic.mathdoc.fr/item/DVMG_2013_13_2_a1/

[1] Yu. A. Brudnyi, N. Ya.Kruglyak, “Funktory veschestvennoi interpolyatsii”, DAN SSSR, 256:1 (1981), 14–17 | MR

[2] V. I. Ovchinnikov, “The method of orbit in interpolation theory”, Math. Rept., 1, Part 2, Harwood Acad. Publ., London, 1984, 349–515 | MR

[3] M. Cwikel, B. Jawerth, M. Milman, “On the fundamental lemma of interpolation theory”, J. Approx. Theory, 60:1 (1990), 70–82 | DOI | MR | Zbl

[4] A. A. Dmitriev, “O konstante K-delimosti funktsionala Petre pary banakhovykh reshetok”, DV shkola-seminar im. akad. E. V. Zolotova. Vladivostok, 27.08–02.09, Dalnauka, Vladivostok, 2000, 37–38

[5] M. Cwikel, “The K-divisibility constant for couples of Banach lattices”, J. Approx. Theory, 124:1 (2003), 124–136 | DOI | MR | Zbl

[6] I. Berg, I. Lëfstrëm, Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980, 264 pp. | MR

[7] Yu. A. Brudnyi, S. G. Krein, E. M. Semenov, “Interpolyatsiya lineinykh operatorov”, Itogi nauki i tekhniki. Ser. Matem. analiz, 24, VINITI, M., 1986, 3–163 | MR

[8] K. I. Oskolkov, “Approksimativnye svoistva summiruemykh funktsii na mnozhestvakh polnoi mery”, Matem. sb., 103(145):4(8) (1977), 563–589 | MR | Zbl

[9] S. Janson, “Minimal and maximal method of interpolation”, J. Func. Anal., 44 (1981), 50–73 | DOI | MR | Zbl

[10] M. Cwikel, I. Kozlov, “Interpolation of weighted $L^1$ spaces – a new proof of the Sedaev–Semenov theorem”, Illinois J. Math., 46 (2002), 405–419 | MR | Zbl