2 INTRODUCTION

we mean by this that there exists constants A, B , dependin g only on p an d

n, suc h that

(0.3') A\\f\\ p\\gl(f)\\pB\\f\\p

for al l / € L

p(Rn).

Th e expression g x(f) i s a version o f th e Littlewood -

Paley ^-function .

The inequalities (0.3' ) ca n be used to give us a characterization of L

p(Rn)

in terms of harmonic functions on R"

+1.

Th e expression on the right in (0.2)

makes sense for any such function u(x, t) (independentl y of whether it arises

from an f €

Lp

a s in (0.1)). The n it can be shown (if we assume u satisfie s

u(x, i) -• 0 a s t -* oo) tha t the condition

for 1 p o o implie s that u ha s the form (0.1) wit h f e L

p

and , con-

sequently, (0.3 ) i s true . I n a simila r fashio n eac h o f th e functio n space s

mentioned above (H

p,

L

p

k

, etc.) i s characterized by a mixed norm estimate

in R"

+1o

n the derivatives of appropriate order of u(x , t) (th e term "mixed

norm" refers to the use of an L

2

nor m in the vertical /-directio n and an L

p

norm in the space variable x £ R

n).

Originally, properties of / (suc h as / € L

p,

p 1) wer e used to obtain

information abou t th e solutio n u(x , t) o f th e Dirichle t proble m (suc h a s

«(•, t) €

Lp(Rn)

fo r each t 0 o r the fact that u satisfies (0.4)). Littlewood -

Paley theory turns this around by using characterizations, such as (0.3), of the

norm of / i n terms of the solution u , a s a means of studying the functio n

space to which / belongs . Al l this will be discussed in detail in the following

lectures, where references will also be given.

There are, or course, other points of view that lead to unified approache s

in the study of these spaces. Perhap s the most natural concept that one might

consider for this purpose is that of a basis. Th e spaces we have mentione d

are all "function spaces" (more precisely, we shall see that their elements are

equivalence classe s of tempered distributions). I t is then reasonable to look

for a collection of functions that, in some sense, spans each of these spaces.

In fact, thi s will b e a main them e o f thes e lectures : w e shal l sho w ho w t o

generate all these spaces by certain general spanning sets of functions having

many special features. Th e first clas s of generating functions we construct will

be far from being a linearly independent set and, moreover, it will depend on

the element o f th e function spac e we are considering (thi s is much like the

situation whe n element s o f th e Hard y spac e H

l

ar e expressed i n term s of

atoms). A s we advance through our lectures, however, we will see how these

spanning set s will evolv e int o mor e canonical set s and, eventually , w e shall

obtain actual bases (of wavelets).