## W MU Of T

(with the usual modification if q = oo).

As we have already said, interpolating the couple (Loo(f2), Li(iî)) by the real method we get Lv{fi) if q == p. If q ^ p we get Lorentz spaces. More precisely, for l/p = 6 and 1 < q < oo, we have

(Loo(fi),Li(ii))*l9 = LPM = {/ : ||/|UPi, = (t1/pr*(t))9 jj

Here /* is the non-increasing rearrangement of /

f* (t) = inf{<5 > 0 : |{x€fi: |/(x)| > 6}\ < t} and we have put r*(t) = \J*r(s)dS.

Interpolating with the function parameter Qo^ we get the Lorentz-Zygmund spaces

Namely, for l/p = 9, 1 < q < oo and b £ R, we have

Lorentz-Zygmund spaces are considered in the book by Bennett and Sharpley [2] and in the paper by Bennett and Rudnick [1]. For p = q we recover the Zygmund spaces

Returning to the real interpolation spaces, we always have

Moreover, since A\, we have for 1 < p, q < 00

The next definition is modelled in Theorem 2.1.

Definition 3.1. Let Aq and Ai be two Banach spaces with Aq A\. Let 0 < 6 < 1 and let j0 = j0{6) € N such that, for all j e N with j > j0, crj = 6 + 2~j < 1 and Aj=0- 2~j > 0.

(i) Assume b < 0. Then ^¿(^(log is the collection of all a € a<7)<1 AVtq which have a finite norm

j=jo

(with the usual modification if q = 00).

(ii) Let b > 0. Then A6iq(\ogA)b is the collection of all a € A\ which can be represented as