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On an Extreme Class of Real Interpolation Spaces

Cobos, Fernando and Fernandez-Cabrera , Luz and Kuehn, Thomas and Ullrich, Tino (2009) On an Extreme Class of Real Interpolation Spaces. Journal of Functional Analysis , 256 (7). pp. 2321-2366. ISSN 0022-1236

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Abstract

We investigate the limit class of interpolation spaces that comes up by the choice θ = 0 in the definition of the real method. These spaces arise naturally interpolating by the J -method associated to the unit square. Their duals coincide with the other extreme spaces obtained by the choice θ = 1. We also study the behavior of compact operators under these two extreme interpolation methods. Moreover, we establish some interpolation formulae for function spaces and for spaces of operators.


Item Type:Article
Uncontrolled Keywords:Extreme interpolation spaces; Real interpolation; J -functional; K-functional; Interpolation methods; Compact-Operators; Banach-Spaces; Polygons; Extrapolation; Reiteration; Duality; Mathematics associated to polygons; Compact operators; Lorentz–Zygmund function spaces; Spaces of operators
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:14957
References:

[1] W.O. Amrein, A. Boutet de Monvel, V. Georgescu, C0-Groups, Commutators Methods and Spectral Theory of N-Body Hamiltonians, Progr. Math., vol. 135, Birkhäuser, Basel, 1996.

[2] B. Beauzamy, Espaces d’Interpolation Réels: Topologie et Géométrie, Lecture Notes in Math., vol. 666, Springer, Heidelberg, 1978.

[3] C. Bennett, K. Rudnick, On Lorentz–Zygmund spaces, Dissertationes Math. 175 (1980) 1–67.

[4] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.

[5] J. Bergh, J. Löfström, Interpolation Spaces, an Introduction, Springer, Berlin, 1976.

[6] P.L. Butzer, H. Berens, Semi-Groups of Operators and Approximation, Springer, New York, 1967.

[7] F. Cobos, D.E. Edmunds, A.J.B. Potter, Real interpolation and compact linear operators, J. Funct. Anal. 88 (1990) 351–365.

[8] F. Cobos, D.L. Fernandez, On interpolation of compact operators, Ark. Mat. 27 (1989) 211–217.

[9] F. Cobos, L.M. Fernández-Cabrera, A. Manzano, A. Martínez, Real interpolation and closed operator ideals, J. Math. Pures Appl. 83 (2004) 417–432.

[10] F. Cobos, L.M. Fernández-Cabrera, A. Manzano, A. Martínez, Logarithmic interpolation spaces between quasi- Banach spaces, Z. Anal. Anwend. 26 (2007) 65–86.

[11] F. Cobos, L.M. Fernández-Cabrera, J. Martín, Some reiteration results for interpolation methods defined by means of polygons, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 1179–1195.

[12] F. Cobos, L.M. Fernández-Cabrera, H. Triebel, Abstract and concrete logarithmic interpolation spaces, J. Lond. Math. Soc. 70 (2004) 231–243.

[13] F. Cobos, P. Fernández-Martínez, A duality theorem for interpolation methods associated to polygons, Proc. Amer. Math. Soc. 121 (1994) 1093–1101.

[14] F. Cobos, P. Fernández-Martínez, A. Martínez, Y. Raynaud, On duality between K- and J -spaces, Proc. Edinb. Math. Soc. 42 (1999) 43–63.

[15] F. Cobos, T. Kühn, T. Schonbeck, One-sided compactness results for Aronszajn–Gagliardo functors, J. Funct. Anal. 106 (1992) 274–313.

[16] F. Cobos, J. Martín, On interpolation of function spaces by methods defined by means of polygons, J. Approx. Theory 132 (2005) 182–203.

[17] F. Cobos, J. Peetre, Interpolation of compact operators: The multidimensional case, Proc. Lond. Math. Soc. 63 (1991) 371–400.

[18] A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994.

[19] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992) 333– 343.

[20] D.E. Edmunds, W.D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer, Berlin, 2004.

[21] D.E. Edmunds, H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996.

[22] S. Ericsson, Certain reiteration and equivalence results for the Cobos–Peetre polygon interpolation method, Math. Scand. 85 (1999) 301–319.

[23] D.L. Fernandez, Interpolation of 2n Banach spaces, Studia Math. 45 (1979) 175–201.

[24] L.M. Fernández-Cabrera, A. Martínez, Interpolation methods defined by means of polygons and compact operators, Proc. Edinb. Math. Soc. 50 (2007) 653–671.

[25] A. Gogatishvili, B. Opic,W. Trebels, Limiting reiteration for real interpolation with slowly varying functions,Math. Nachr. 278 (2005) 86–107.

[26] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, RI, 1969.

[27] M.E. Gomez, M. Milman, Extrapolation spaces and almost-everywhere convergence of singular integrals, J. Lond. Math. Soc. 34 (1986) 305–316.

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