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