By Elias M. Stein

The aim of this booklet is to explain a definite variety of effects related to the examine of non-linear analytic dependence of a few functionals bobbing up evidently in P.D.E. or operator concept.

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**Extra resources for Beijing lectures in harmonic analysis**

**Example text**

The proof is a 1. /J space valued functions. Fix x 2 . Then f LP simple iteration of the one-parameter case given previously. Pdx 2 1 R define -Qff(x l ,x 2) f '(x ")(,, ",(x)' = p[f], f 2(x)g2(x)dx + 1 d,' dX} . /Jt(x) = Qt(g)(x), Q~, i = 1,2, will denote the operator acting in the i th variable. 2) 1 2 2 'VI 'V2 2 dt 1dt 2 [Qt Qt f(x 1 ,x 2)] [Pt P t g(x 1,x 2 )] - t-tdx 1dx 2 . 3) R 2 XR 2 + + + + f-J ff J" JJ s xl fR x 2 ,t 2 x 1 ,t 2 + + IIII if f x2 t2 Now I 1= I dt dx [Qf f(x)]2[Pf g(x)]2dx 1 _ 2_ 1 = I + II.

Let us begin by giving the most basic example, which dates back to I II II II Jessen, Marcinkiewicz, and Zygmund. We are referring to a maximal opera tor on R n which commutes with the full n-parameter group of dilations Mcn)f(x) = sup II xfR II I THEOREM OF JESSEN_MARCINKIEWICZ-ZYGMUND functions f(x) in the unit cube of R (xl' x 2' "', x n) ~ (01 Xl '02 x 2' ... , 0nxn), where 0i > 0 is arbitrary. This is the "strong maximal operator," Mcn), defined by II -l.. R! f If(t)1 dt >al! ~ ~ we have ~ l1 f l L (lOg L)n-l CQo )· n "" 2, which is already entirely typical.

_;;,,,,,,,- -- 76 .... :: ... ~.. W1g;::::<':"~~'<-~~:~~Jt' ROBERT FEFFERMAN MULTIPARAMETER FOURIER ANALYSIS WI~I (,b, ~ W- I(Pl)t s c . l (*) Conversely, assume w f AP. Q (fa-l)da):s C f Q~ R J n MP(fa)da a Iflda. ~l so that J J- ~ J