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 9 t) = (P t *f)(x), where P t (x) = c n t(\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
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