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

, /-!=,