RT Journal Article T1 Logarithmic Interpolation Spaces Between Quasi-Banach Spaces A1 Cobos, Fernando A1 Fernández-Cabrera, Luz M. A1 Manzano, Antonio A1 Martinez, Antón AB Let A0 and A1 be quasi-Banach spaces with A0 ,! A1. By means of a direct approach, we show that the interpolation spaces on (A0;A1) generated by the function parameter tµ(1 + j log tj)¡b can be expressed in terms of classical real inter-polation spaces. Applications are given to Zygmund spaces Lp(log L)b(­), Lorentz-Zygmund function spaces and operator spaces de¯ned by using approximation num-bers. PB Heldermann Verlag SN 0232-2064 YR 2007 FD 2007 LK https://hdl.handle.net/20.500.14352/49870 UL https://hdl.handle.net/20.500.14352/49870 LA eng NO [1] Arazy, J., Fisher, S. D. and Peetre, J., Hankel operators on weighted Bergman spaces. Amer. J. Math. 110 (1988), 989 { 1053. [2] Bennett, C. and Rudnick, K., On Lorentz-Zygmund spaces. Warsaw: Polish Acad. Sci. Inst. Math., Dissertationes Math. 175 (1980). [3] Bennett, C. and Sharpley, R., Interpolation of Operators. Boston: Academic Press 1988. [4] Bergh, J. and LÄofstrÄom, J., Interpolation Spaces. An Introduction. Berlin: Springer 1976. [5] Cobos, F., On the Lorentz-Marcinkiewicz operator ideal. Math. Nachr. 126 (1986), 281 { 300. [6] Cobos, F., Entropy and Lorentz-Marcinkiewicz operator ideals. Arkiv Mat. 25 (1987), 211 { 219. [7] Cobos, F., Fern¶andez-Cabrera, L. M. and Triebel, H., Abstract and concrete logarithmic interpolation spaces. J. London Math. Soc. 70 (2004), 231 { 243. [8] Edmunds, D. E. and Evans, W. D., Hardy Operators, Function Spaces and Embeddings. Heidelberg: Springer 2004. [9] Edmunds, D. E. and Triebel, H., Logarithmic Sobolev spaces and their appli- cations to spectral theory. Proc. London Math. Soc. 71 (1995), 333 { 371. [10] Edmunds, D. E. and Triebel, H., Function Spaces, Entropy Numbers, Di®er- ential Operators. Cambridge: Cambridge University Press 1996. [11] Edmunds, D. E. and Triebel, H., Logarithmic spaces and related trace prob- lems. Funct. Approx. Comment. Math. 26 (1998), 189 { 204. [12] Fern¶andez-Cabrera, L. M., Cobos, F., Hern¶andez, F. L. and S¶anchez, V. M., Indices de¯ned by interpolation scales and applications. Proc. Royal Soc. Ed- inburgh 134A (2004), 695 { 717. [13] Fiorenza, A. and Karadzhov, G. E., Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwendungen 23 (2004), 657 { 681. [14] Gohberg, I. C. and Krein, M. G., Introduction to the Theory of Linear Non- selfadjoint Operators (transl. from Russian). Providence, R.I.: Amer. Math. Soc. 1969. [15] Gohberg, I. C. and Krein, M. G., Theory and Applications of Volterra Operators in Hilbert Space (transl. from Russian). Providence, R.I.: Amer. Math. Soc. 1970. [16] Gustavsson, J., A function parameter in connection with interpolation of Ba- nach spaces. Math. Scand. 42 (1978), 289 { 305. [17] Janson, S., Minimal and maximal methods of interpolation. J. Funct. Anal. 44 (1981), 50 { 73. [18] Jawerth, B. and Milman, M., Extrapolation Theory with Applications. Provi- dence, R.I.: Mem. Amer. Math. Soc. 89 (1991), no. 440. 86 F. Cobos et al. [19] Karadzhov, G. E. and Milman, M., Extrapolation theory: new results and applications. J. Approx. Theory 133 (2005), 38 { 99. [20] KÄonig, H., Interpolation of operator ideals with an application to eigenvalue distribution problems. Math. Ann. 233 (1978), 35 { 48. [21] KÄonig, H., Eigenvalue Distribution of Compact Operators. Basel: BirkhÄauser 1986. [22] Milman, M., Extrapolation and Optimal Decompositions. Lecture Notes in Mathematics 1580. Berlin: Springer 1994. [23] Opic, B. and Pick, L., On generalized Lorentz-Zygmund spaces. Math. Inequal. Appl. 2 (1999), 391 { 467. [24] Peetre, J., A Theory of Interpolation of Normed Spaces. Notas Mat. 39 (Lec- tures Notes, Brasilia, 1963). Rio de Janeiro: Inst. Mat. Pura Apl. 1968. [25] Persson, L.-E., Interpolation with a parameter function. Math. Scand. 59 (1986), 199 { 222. [26] Pietsch, A., Operator Ideals. Amsterdam: North-Holland 1980 [27] Pietsch, A., Eigenvalues and s-Numbers. Cambridge: Cambridge University Press 1987. [28] Stein, E. M., Singular Integrals and Di®erentiability Properties of Functions. Princeton, N.J.: Princeton Univ. Press 1970. [29] Strichartz, R. S., A note on Trudinger's extension of Sobolev's inequality. In- diana Univ. Math. J. 21 (1972), 841 { 842. [30] Triebel, H., Interpolation Theory, Function Spaces, Di®erential Operators. Amsterdam: North-Holland 1978 (sec. ed. Leipzig: Barth 1995.) [31] Triebel, H., Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces. Proc. London Math. Soc. 66 (1993), 589 { 618. [32] Triebel, H., Fractals and Spectra Related to Fourier Analysis and Function Spaces. Basel: BirkhÄauser 1997. [33] Trudinger, N. S., On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473 { 483. NO Spanish Ministerio de Educación y Ciencia DS Docta Complutense RD 16 may 2024