Calderón-Zygmund Operators, Pseudo-Differential Operators by Jean-Lin Journé (auth.)

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F, If] < n = nsgnf, Ill > n Then we can obtain the appropriate and take the limit as n tends to infinity. This completes the proof. We should remark that we have really proved a bit more. ceeding calculation shows that for all (3) "Gk , and I{xEQ : fs(x)>%+~ ~ , f In particular, the pre- positive, < Y~}I =< 2nyIQI for each maximal dyadic cube Indeed, let Q0 Q in ~ . _*,# # * ~t6) 8 <__ C(fs) 6 . %0 > inf f6 ° Consider the Q0 maximal dyadic cubes in that it intersects. ~ for ~ =40+ ~ . ) If we apply (3) above and sum over all such maximal dyadic cubes that also happen to be included in I{xEQo: QO ~ then we obtain that (f6(x)-~0 ) > 2~ , f~(x) ~][ For 7> 0 small enough, we may apply the good (~n, dx) true when is replaced by A's inequality (which is still or any measure space, for that matter) (Qo~dx) , to conclude that max(f~-AO, 0)dx <= C ~ f~ dx , QO QO W with C independent of ~0 " If we let Q~ k0 tend to - inf f 6 ~ ) I d x < 0 y E QO / inf f6(x) , xEQ then we obtain C ~ f~dx , = QO _*,# #* ( ~ ) ~ ~ C(f6) 6 which is clearly stronger than This inequality also holds for the non-dyadic versions of these operators.

See section II of chapter 0) is useful for proving weighted norm inequalities. [ f 6=> X ~/=y(w,p)>0 where ~p such that satisfies 2P~p~]I will be effected through a decomposition of cubes, the characterization of A= weights forall [f~ > ~} wEA cube ~ = {f~ > ~} , we obtain Q , where into maximal disjoint ? (dyadic) small enough. , apply the aforementioned C(w) > i . Because the proof (see the second lenTna of section II in chapter 2) implies that (i) follows from (2), for Let ~>0 lenmm) These two facts imply that fELP(~dx) .

The Hilbert transform and similar Motivations for these generaliza- tions can be found in [F]. e. if ) . f Under these assumptions~ is in some LP(Rn, dx) To study the LP-boundedness p=2 , for it turns out that the above limit p E [ij + m [ of such operators, one begins by studying which can be handled via Plancherel' s theorem. the case p=2 and from the properties . of the kernel The case k , p~2 the case follows from defined by k(x,y) = ~ i x - Y ~ i ~IIx - yllJ llx-ylln Thus, in k is defined on the complement of the diagonal, A = [(x,y) 6 • a × R n / x = y } , R nx R n .