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" +1 o 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).
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