## Beijing lectures in harmonic analysis by Elias M. Stein

By Elias M. Stein

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

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HP spaces - one and several parameters In this lecture we wish to discuss another chapter of harmonic analysis relating to differentiation theory and singular integrals, namely Me is of weak type (p/2)'. I To estimate

Now M is bounded on I LIla so that M(h)fL l/a and so G*(L I / a . ltfollowsthat F*fL . Just as for a random f (LI(Rn) we do not necessarily have Rif (LI(Rn) (singular integrals do not preserve L 1 ) it is also not true that for an I arbitrary L 1 function f that for u = P[f), u* (L . But if f (HI(R~+I) then u* (LI(R n). Thus the nontangential maximal function F*(x) = sup \F(y,t)\ l n L\R ) (y,t)lr(X) there, u is given as an average of its boundary values according to the Poisson integra 1: S C for all t > 0) if and only if the analytic function f (HI(R~+I).

4) O f -00 IF(x +it)ldx 'V IluliL 1 CR 1) + IlvilL 1 CR 1) . 1. t' So we may view the space H1 through its boundary values as the space of a 11 real valued functions f ( L 1(R 1) whose Hilbert transforms are L 1 as well. 'r . "".. " -.. ~_a3t . - ....... _. ----- ---_~~=,-_. MULTIPARAMETER FOURIER ANALYSIS ROBERT FEFFERMAN cnt u(x,t) =f * Pt(x); f(x) =u(x,O) and Pt(x) = (lxI 2 +t 2 )(n+l)/2 If we want a theory of HP(Rn) then, following Stein and Weiss we may consider the functions F(x,t) in R~+l = l(x,t)lx (R n , t >01 whose values lie in R n+ l : F(x,t) = (uo(x,t), ...