SOME REMARKS ON THE FEFFERMAN-STEIN INEQUALITY 1 ...

16 ANDREI K. LERNERLet T be a Calderón-Zygmund singular integral operator, that is,T = p.v.f ∗ K with kernel K satisfying the standard conditions∥ ̂K∥ L ∞ ≤ c, |K(x)| ≤ c/|x| n ,|K(x) − K(x − y)| ≤ c|y|/|x| n+1 for |y| < |x|/2.Actually, the results described below hold for more general Calderón-Zygmund operators as well.The weighted theory of the Fefferman-Stein inequality has been developedin parallel to the one of Coifman’s inequality relating singularintegrals and the maximal function. Namely, it was proved by Coifman[3] (see also [4]) that if w ∈ A ∞ , then for any appropriate f,(6.3) ∥T f∥ L p (w) ≤ c∥Mf∥ L p (w) (1 < p < ∞).This result was based on a good-λ inequality related T f and Mf.However, the Fefferman-Stein inequality originally was also proved withthe help of a good-λ inequality related f and f # . Therefore, it hasbeen quickly realized that if w ∈ A ∞ , then (5.1) holds. After that,Sawyer [24] observed that the weak A ∞ condition is enough for (6.3).The same argument applies to (5.1).In [22], Muckenhoupt established that in the case when T is theHilbert transform, the C p condition is necessary for (6.3), and he conjecturedthat C p is also sufficient. Note that this question is still open.In [25], Sawyer proved that if ε > 0, then the C p+ε condition is sufficientfor (6.3). Using almost the same arguments, Yabuta [28] showedthat C p is necessary for (5.1) and C p+ε is sufficient.Here we give a completely different proof of a slightly improved versionof Yabuta’s result. Given p > 1, let φ p be a non-decreasing,doubling (i.e., φ p (2t) ≤ cφ p (t)) function on (0, 1) satisfying∫ 10φ p (t) dt < ∞.tp+1 We say that a weight w satisfies the ˜C p condition if there are positiveconstants c, δ such that for any cube Q and any subset E ⊂ Q,w(E) ≤ c(|E|/|Q|) δ ∫R n φ p (Mχ Q )w.Theorem 6.1. The C p condition is necessary for∫∫(6.4)M p,r (f, w)|f| dx ≤ c (Mf) p w dx,R n R nand the ˜C p is sufficient.