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
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
in terms of harmonic functions on R"
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
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
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
, etc.) i s characterized by a mixed norm estimate
in R"
n the derivatives of appropriate order of u(x , t) (th e term "mixed
norm" refers to the use of an L
nor m in the vertical /-directio n and an L
norm in the space variable x £ R
Originally, properties of / (suc h as / L
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)
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
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).
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