By Elias M. Stein

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

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**Example text**

We claim ~ 86 ~ ~ ~ ~ ~ ' I I l _ ROBERT FEFFERMAN on L(p/2)«p/2)' is the exponent dual to p/2). More pre cisely, suppose that TO is bounded on LP(R 2 ) and we assume the Me I{MeXE > ~fl s C1EI. \1 e [R n [U I (:£ Ifk l2) i j j D ! i l i ~ 00, 1/2 IILP(R2) s cll (:£lf k \2) -·iTk· x p iTk' x R·]I J < 21. jR I. <);:;, 21. '(j s i{Me

Q. f definition of Mil we insist that the balls be centered at x then Mil is We should also remark that the original proof of the weight norm inequali ties for M on R n made use of Mil as well. In fact, if w f AP it is not e>k J J given Lebesgue measure by another measure dll. on a refinement of the usual Vitali covering lemma due to Besocovitch. J E~ = Q~ - u Q~ then choosing C large enough insures that J own right, since it is natural to ask what happens if we replace the Gcx:I In fact, if Il is any measure finite on compact sets and if, in the So far we have used only arithmetic.

Dxn_lS k 1 If¢~cI IEkllS'kl - f¢~ k So k S'k (by induction) we have IURjl ~ CIU(Rk)d l ~ C' I IRkl ~ C'IURkl