On an Extreme Class of Real Interpolation Spaces

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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.
[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.