CHAPTER 1
A Class of Function Spaces
1.1. Definitions and Basic Properties
Everywhere in the following the letter E denotes a quasi-Banach space of se-
quences of Lebesgue measurable functions on R^, which has the lattice property
with respect to the natural ordering.
More precisely this means that there is a non-negative function || || on E
(a quasi-norm), which has the same properties as a norm, except for the triangle
inequality, and in addition satisfies the following conditions:
(i) The metric space (E, || ||) is complete.
(ii) If {fi}iZ0 G E, and {^}^
0
is a sequence of measurable function such
that \gi\ \fi\ a.e. for all i, it follows that {^}^
0
£
E1,
and
||{fc}£obll{/ih~ob-
(iii) There is a constant CE 1 such that
(1.1.1) \\F + G\\ECE{\\F\\E + \\G\\E)
for all F,GeE.
It is a well-known property of quasi-norms that (iii) can be replaced by the
following:
(iii7) There exist constants 0 K 1, and C'E 1, such that for any family
{Fi}Ji=0 of elements in E one has the inequality
Here
K
can be chosen so that 0 n ^o, where
(2CE)2K°
2, and then C'E =
(2CE)2K-
The property (iii') was found, apparently independently, by Tosio Aoki [6] and
S. Rolewicz [44]; see also Rolewicz [45], Th. 3.2.1. It is also proved in many other
places, see e.g. Bergh-Lofstrom [7], Lemma 3.10.1, DeVore-Lorentz [15], Ch. 2,
Th. 1.1, or J. Heinonen [25], Prop. 14.5. The ranges of K and C'E given above come
from [25].
The following definition will play a central role in what follows.
DEFINITION 1.1.1. Let £+, e- e R, 0 r oc, and d 0. We define a
class £(£+,£_, r, d) of spaces E by saying that E G £(£+,£_, r, d) if the following
conditions are satisfied:
(a) The linear operators 5+ (left shift) and 5_ (right shift), defined by
s+({fi}^o) = {.A+iL^o
5-({/
i

0
)={/
i
-i}£
0
, /-!=,
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