# On equivalence of K- and J-methods for (n+1)-tuples of Banach spaces

Irina Asekritova; Natan Krugljak

Studia Mathematica (1997)

- Volume: 122, Issue: 2, page 99-116
- ISSN: 0039-3223

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topAsekritova, Irina, and Krugljak, Natan. "On equivalence of K- and J-methods for (n+1)-tuples of Banach spaces." Studia Mathematica 122.2 (1997): 99-116. <http://eudml.org/doc/216371>.

@article{Asekritova1997,

abstract = {It is shown that the main results of the theory of real interpolation, i.e. the equivalence and reiteration theorems, can be extended from couples to a class of (n+1)-tuples of Banach spaces, which includes (n+1)-tuples of Banach function lattices, Sobolev and Besov spaces. As an application of our results, it is shown that Lions' problem on interpolation of subspaces and Semenov's problem on interpolation of subcouples have positive solutions when all spaces are Banach function lattices or their retracts. In general, these problems have negative solutions.},

author = {Asekritova, Irina, Krugljak, Natan},

journal = {Studia Mathematica},

keywords = {real interpolation; reiteration theorems; Banach function lattices; Sobolev and Besov spaces; Lions' problem; interpolation of subspaces; Semenov's problem},

language = {eng},

number = {2},

pages = {99-116},

title = {On equivalence of K- and J-methods for (n+1)-tuples of Banach spaces},

url = {http://eudml.org/doc/216371},

volume = {122},

year = {1997},

}

TY - JOUR

AU - Asekritova, Irina

AU - Krugljak, Natan

TI - On equivalence of K- and J-methods for (n+1)-tuples of Banach spaces

JO - Studia Mathematica

PY - 1997

VL - 122

IS - 2

SP - 99

EP - 116

AB - It is shown that the main results of the theory of real interpolation, i.e. the equivalence and reiteration theorems, can be extended from couples to a class of (n+1)-tuples of Banach spaces, which includes (n+1)-tuples of Banach function lattices, Sobolev and Besov spaces. As an application of our results, it is shown that Lions' problem on interpolation of subspaces and Semenov's problem on interpolation of subcouples have positive solutions when all spaces are Banach function lattices or their retracts. In general, these problems have negative solutions.

LA - eng

KW - real interpolation; reiteration theorems; Banach function lattices; Sobolev and Besov spaces; Lions' problem; interpolation of subspaces; Semenov's problem

UR - http://eudml.org/doc/216371

ER -

## References

top- [A1] I. U. Asekritova, On the weak K-divisibility of (n+1)-tuples of ideal Banach spaces, in: Studies in the Theory of Functions of Several Real Variables, Yaroslavl', 1990, 15-21 (in Russian).
- [A2] I. U. Asekritova, A real interpolation method for finite collections of Banach spaces, in: Studies in the Theory of Functions of Several Real Variables, Yaroslavl', 1981, 9-17 (in Russian).
- [A3] I. U. Asekritova, On the property of K-divisibility for finite collections of Banach spaces, in: Studies in the Theory of Functions of Several Real Variables, Yaroslavl', 1984, 3-9 (in Russian).
- [A4] I. U. Asekritova, A counterexample to K-divisibility for (n+1)-tuples of Banach spaces, in: Studies in the Theory of Functions of Several Real Variables, Yaroslavl', 1988, 5-15 (in Russian).
- [A5] I. U. Asekritova, Theorem of reiteration and K-divisionable (n+1)-tuples of Banach spaces, Funct. Approx. Comment. Math. 20 (1992), 171-175.
- [BL] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. Zbl0344.46071
- [BK] Yu. A. Brudnyĭ and N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces 1, North-Holland, Amsterdam, 1991.
- [CNPP] M. J. Carro, L. I. Nicolova, J. Peetre and L.-E. Persson, Some real interpolation methods for families of Banach spaces. A comparison, research report 22 (1994), Department of Mathematics, Lulea University of Technology.
- [CP] F. Cobos and J. Peetre, Interpolation of compact operators: the multidimensional case, Proc. London Math. Soc. 63 (1991), 371-400. Zbl0702.46047
- [CJ] M. Cwikel and S. Janson, Real and complex interpolation methods for finite and infinite families of Banach spaces, Adv. in Math. 66 (1987), 234-290. Zbl0646.46070
- [CN] M. Cwikel and P. Nilsson, Interpolation of weighted Banach lattices, Technion Preprint Series 834 (1989).
- [JNP] S. Janson, P. Nilsson and J. Peetre, Notes on Wolff's note on interpolation spaces, Proc. London Math. Soc. 48 (1984), 283-289. Zbl0532.46046
- [KPS] S. G. Kreĭn, Yu. I. Petunin and E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English transl.: Amer. Math. Soc., Providence, 1982.
- [L] J. L. Lions, Problems in interpolation of operators and applications (Problem list of the Special Session on Interpolation of Operators and Applications), Notices Amer. Math. Soc. 22 (1975), 126.
- [LM] J. L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968. Zbl0165.10801
- [M] L. Maligranda, On commutativity of interpolation with intersection, Suppl. Rend. Circ. Mat. Palermo 10 (1985), 113-118. Zbl0616.41001
- [S] G. Sparr, Interpolation of several Banach spaces, Ann. Mat. Pura Appl. 99 (1974), 247-316. Zbl0282.46022
- [W] R. Wallsten, Remarks on interpolation of subspaces, in: Function Spaces and Applications (Proc., Lund 1986), Lecture Notes in Math. 1302, Springer, 1988, 410-419.
- [Y] A. Yoshikawa, Sur la théorie d'espaces d'interpolation - les espaces de moyenne de plusieurs espaces de Banach, J. Fac. Sci. Univ. Tokyo Sect. 1 Math. 16 (1970), 407-468. Zbl0197.09901

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