## Info

(with the usual modification if <7 = 00) is finite.

If b = 0 we get the usual Besov spaces Bpg(R"). Besov spaces of generalized smoothness are Banach spaces and the norms are equivalent to each other for admissible choices of <p. These spaces attracted some attention in connection with fractal analysis and related spectral theory. A short description can be found in Triebel's book . A detailed study of these spaces is made in the paper by Moura . Let

with l<p<oo,l<Q<oo, and —00 < si < s0 < 00. Then Ao <-> j4i. Moreover (see  and )

B<^(R") = (B^(Rn),B^(Rn))geb;g where s = (1 — 9)so + Osi. Consequently, using Theorem 3.1 we derive:

Corollary 4.2. Let 0 < 6 < 1,1 < q < 00, b e R, and let 1 < p < 00, —00 < si < sq < 00 and s = (1 — 6)sq + 6s\. (i) Let b< 0. Then

where the infimum is taken over all representations oo

j=jo

The last application refers to operator spaces defined by summability conditions on the singular numbers. Let if be a Hilbert space and given any bounded linear operator T £ £(H), let {s„(T)} be the sequence of the singular numbers of T, defined by sn(T) = inf{||T — R\\ : rankiî < n} , n £ N.

For l<p<oo, l<ç<oo and 6 £ t, the Lorentz-Zygmund operator space CPtqtb{H) consists of all T £ C(H) having a finite norm oo n , , rP,g,b(T)=(E(ni~1( l + logny^CT))^-1) '

(with the usual modification if q = oo).

The space CPtgtb(H) is the component over H of the Lorentz-Zygmund operator ideal that has been studied in ,  and . Note that T belongs to CPiqib(H) if and only if {sn(T)} belongs to the Lorentz-Zygmund sequence space £p,q(log£)b- For b = 0 we recover the Lorentz operator space (Cpiq(H),Tptq), and for b = 0 and p = q,we get the Schatten p—class (.Cp(H),Tp) (see ,  and ).

In order to apply Theorem 3.1 in this context, we put Aq = C\{H) and A\ = C(H). Then Aq c—> A\ and, according to [5, Thm. 5.1], we have

Theorem 3.1 yields:

Corollary 4.3. Let H be a Hilbert space. Let 1 < p < oo, 1 < q < oo, and let jo = jo(p) £ N such that for all j £ N with j > jo,

pUj p pV3 p

(i) Let b < 0. Then CPtq<b{H) is the set of all T £ C{H) such that oo \ «

(equivalent norms).

(ii) Let b > 0. Then Cp>qfi{H) consists of all T € C(H) which can be represented as T = Y^jLj0 Tj with Tj e Cp^j ¡q{H) such that i oo \ 9

Furthermore, the infimum over the expression in (9) is an equivalent norm in £Pigib(H).

Remark 4.1. All results we have shown refer to Banach spaces, but logarithmic interpolation spaces can also be considered in the class of quasi-Banach spaces. Details can be found in the joint paper by Fernández-Cabrera, Manzano, Martínez and the present author . Among other things, we establish there that statement (ii) in Theorem 2.1 holds for 0 < p < 1 as well, and we give applications to operator spaces on Banach spaces.

References

1. C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980) 1-67.

2. C. Bennett and R. Sharpley, "Interpolation of operators", Academic Press, Boston, 1988.

3. J. Bergh and J. Lófstróm, "Interpolation spaces. An introduction", Springer, Berlin, 1976.

4. B. Carl and H. Triebel, Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spaces, Math. Ann. 251 (1980) 129-133.

5. F. Cobos, On the Lorentz-Marcinkiewicz operator ideal, Math. Nachr. 126

6. F. Cobos, Entropy and Lorentz-Marcinkiewicz operator ideals, Arkiv Mat. 25

7. F. Cobos, Duality and Lorentz-Marcinkiewicz operator spaces, Math. Scand. 63 (1988) 261-267.

8. F. Cobos and D.L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, in Lecture Notes in Mathematics 1302, Springer, Berlin, 1988, pp.158-170.

9. F. Cobos, L. M. Fernández-Cabrera, A. Manzano and A. Martinez Logarithmic interpolation spaces between quasi-Banach spaces, preprint 2004.

10. F. Cobos, L. M. Fernández-Cabrera and H. Triebel, Abstract and concrete logarithmic interpolation spaces, J. London Math. Soc. 70 (2004) 231-243.

11. D. E. Edmunds and W.D. Evans, "Hardy Operators, Function Spaces and Embeddings", Springer Monograps in Mathematics, Berlin, 2004.

12. D. E. Edmunds and H. Triebel, Eigenvalue distributions of some degenerate elliptic operators: an approach via entropy numbers, Math. Ann. 299 (1994) 311-340.

13. D. E. Edmunds and H. Triebel, Logarithmic Sobolev spaces and their applications to spectral theory, Proc. London Math. Soc. 71 (1995) 333-371.

14. D. E. Edmunds and H. Triebel, "Function spaces, entropy numbers, differential operators", Cambridge University Press, Cambridge, 1996.

15. D. E. Edmunds and H. Triebel, Logarithmic spaces and related trace problems, Funct. Approx. Comment. Math. 26 (1998) 189-204.

16. I. C. Gohberg and M. G. Krein, "Introduction to the theory of linear non-self adjoint operators", American Mathematical Society, Providence, R.I., 1969.

17. J. Gustavsson, A function parameter in connection with interpolation of Ba-nach spaces, Math. Scand. 42 (1978) 289-305.

18. S. Janson, Minimal and maximal methods of interpolation, J. Funct. Anal. 44 (1981) 50-73.

19. B. Jawerth and M. Milman, "Extrapolation theory with applications", Mem. Amer. Math. Soc. 440, Providence, 1991.

20. H. König, "Eigenvalue distribution of compact operators", Birkhäuser, Basel, 1986.

21. C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in Lecture Notes in Mathematics 1070, Springer, Berlin, 1984, pp.183-201.

22. M. Milman, "Extrapolation and optimal decompositions", Lecture Notes in Mathematics 1580, Springer, Berlin, 1994.

23. S. Moura, Function spaces of generalized smoothness, Dissertationes Math. 398 (2001) 1-88.

24. J. Peetre, "A theory of interpolation of normed spaces", Notes Mat. 39 (1968) 1-86. (Lectures Notes, Brasilia, 1963).

25. L.-E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986) 199-222.

26. A. Pietsch, "Eigenvalues and s-numbers", Cambridge University Press, Cambridge, 1987.

27. R. S. Strichaxtz, A note on Trudinger's extension of Sobolev's inequality, Indiana Univ. Math. J. 21 (1972) 841-842.

28. H. Triebel, "Interpolation theory, function spaces, differential operators", North-Holland, Amsterdam, 1978; sec. ed. Barth, Leipzig, 1995.

29. H. Triebel, Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces, Proc. London Math. Soc. 66 (1993) 589-618.

30. H. Triebel, "The structure of functions", Birkhäuser, Basel, 2001.

31. N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-483.