Introduction
Many fields in analysi s requir e th e stud y o f specifi c functio n spaces . I n
harmonic analysi s one soon encounter s the Lebesgue space s if , the Hard y
spaces H
p
, various forms of Lipschitz spaces and the space BMO. Similarly,
the Sobolev spaces L
p
k
ar e basic in the study of partial differential equations .
From th e origina l definition s o f thes e spaces , i t ma y no t appea r tha t the y
are closely related . Ther e are , however, various unified approache s t o thei r
study. I n this monograph we will focus on the Littlewood-Paley theory which
provides one of the most successful unifyin g perspective s on these and othe r
function spaces .
Littlewood-Paley theory arises naturally from consideratio n o f the Dirich-
let problem for the upper half space R"
+1=
{(x , 0 : x e R
n,
t 0} . Given a
function / define d on R
n
, considered as the boundary of R"
+1(b
y the iden-
tification R
n
= {(x, t)\ x e R
n,
/ = 0}) , the solutio n o f th e Dirichlet prob -
lem with boundary value / i s the function u(x, t) tha t is harmonic on R" +1
and equals (in an appropriate sense ) / o n R
n
(tha t is, u(x , 0 ) = f{x)) . If ,
say, feL
p(Rn)9
p 1, the n
(0.1) u(x 9t) = (P t*f)(x),
where P t(x) = c
nt(\x\2
+ f
2)-("+1)/2
?
for t 0 an d x e R
n,
i s the Poisso n
kernel. Man y conclusions follo w easil y from thi s representation o f u(x , t) .
For example , i f / i s bounde d s o i s u; i n fact , \u(x, t)\ H/H ^ sinc e
/ P t(x) dx = 1 fo r all t 0. Littlewood-Pale y theory, however, yields much
more precise information. W e illustrate thi s by introducing th e operator g
{
defined b y
(0.2) g x(f)(x) =
du,
tTt{x.t)
dt\l/2
A basic result of Littlewood-Paley theor y is that
(0.3) \\f\\ p^Ui(f)\\p i f 1 /*;
1
http://dx.doi.org/10.1090/cbms/079/01
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