6 1. A CLASS O F FUNCTION SPACES

are continuous on E, and their norms satisfy for all j G N the inequalities

\\(S+y\\C12-^\ and ||(S_)'|| C2Ve~,

i.e., there are constants C\ and C2, independent of j and {fi}iZ0,

s u c n

that

IKs'_^({/i}s0)[U ^^-iK/imolU -

(b) The operator A^r,d, defined by

Mr,d({fi}£o) = {Mr,dfi}°g0 ,

is bounded on E, i.e.,

||{Mr|d/i}£0|UC'||{/ }£oL-

Set S(e+,£_,r) - U d o 5 ' ( ^ + ^ - ' r ^ ) -

REMARK.

Note that £+ e_, otherwise the space is trivial. In fact, for any

sequence F = {fi}^LQ in E and all j 0

F=(S+)MS_RF),

whence

\\F\\EC1C22-^-£-\

which can be made arbitrarily small if £+ — €- 0.

EXAMPLE.

Let Q denote the class of positive functions

UJ

on (0, 00) with the

property that there is a constant C such that u(x) Cu;(y) whenever | x/y 2.

For w G Q w e denote by VQ the quasi-Banach space of sequences {ai}^l0 such that

the quasi-norm (X^o(la*lu;(22))6,)1//6 is finite (Banach space if 6 1).

The principal examples of spaces E belonging to S(£+,£_,r) are the spaces

Vg(Lp) with 0 p, 9 00 and Lp(l%) with 0 p o o , O 0 o o and UJ e ft, and

with quasi-norms (norms when p, 0 1)

ii{/«}£oii«y(LP) =

II{II/«HMR")OI?

= ( S ^ a w i ^ M * ) ) ' )

1

' ' '

and

11

n m ~

0

i k

w

= ||||{/,(-)}£olLjL

(RJV)

= (E(i^(')i^))')

i / «

2 = 0

L

p

(R^)

One can easily check that these spaces satisfy (1.1.2) with K — min{l,p, 0} and

C'E = 1, that £+ is any constant e such that uj{t)/t£ is C-increasing, and that £_ is

any e such that t£ /uj{t) is C-increasing. (A function /(£) is C-increasing if there is

a constant C such that f(x) Cf(y) whenever x y.)

If cu(t) = tx, —00 A 00, we write 1$ for VQ. Then e+ is any e G (—00, A],

and S- is any £ G [A, +00). Thus, in the most important cases we can choose

£+ = £_. However, these two numbers play different parts in many of the proofs,

so for greater transparency it is useful to keep them notationally different.

It follows from the maximal theorems of Hardy-Littlewood-Wiener (see e.g.

E. M. Stein [48]) and of Fefferman-Stein ([17], see also Stein [49], and K. F.

Andersen and R. T. John [5]) that r can be any number in (0,p) in the case of

IQ(LP),

or in (0,min{p, 0}) in the case of

LP(IQ).

These theorems are basic in the theory of B- and F-spaces, see e.g. the historical

remarks in Triebel [53], Section 2.3.5.