Positivity 6: 17–30, 2002.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Some Extensions of Optimal Interpolation in
Spaces of Lorentz–Zygmund Type
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel.
(Received 23 December 1999; accepted 9 May 2000)
Abstract. The results on optimal interpolation from  are extended to quasinormed spaces with
p<1, to spaces with varying secondary parameters α, E and to spaces of functions deﬁned on the
interval (1, ∞). As a tool for doing this, we construct special mappings which transform these cases
into the basic one, considered in .
Mathematics Subject Classiﬁcation (2000): Primary 46B70, 46E30. Secondary 46E35, 46M35
Key words: Lorentz–Zygmund spaces, optimal interpolation, quasilinear operator, quasinormed
spaces, weak type interpolation
The classical Lorentz–Zygmund spaces L
were introduced and studied
by Bennett and Rudnick in  as spaces of measurable functions f : (0, 1) → R
with the ﬁnite quasinorm
(r < ∞),
(t) (r =∞).
These spaces include as partial cases most of classical spaces of measurable func-
tions studied before, such as Lebesgue spaces L
(when r = p, α = 0), Lorentz
(when α = 0), Zygmund spaces L log L,expL, etc. They were used
for studying quasilinear operators T of joint weak type (a, b; p, q), a < p, b<q,
which in the case of p<∞ is equivalent to two separate weak type conditions
T : L
. One of the main results from  (Theorem C(b), p.
10) can be formulated as follows.
THEOREM 1.1. Every quasilinear operator T of joint weak type (a, b; p, q) with
0 <a<p ∞, 0 <b<q ∞ acts continuously from L
for all s r 1 and all α, β ∈ R such that α +
= β +