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 /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.
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