Logarithmic Interpolation Spaces Between Quasi-Banach Spaces

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Heldermann Verlag
Google Scholar
Research Projects
Organizational Units
Journal Issue
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.
Unesco subjects
[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.