A WEAK TYPE INEQUALITY FOR CAUCHY TRANSFORMS OF FINITE MEASURES

In this note we present a simple proof of a recent result of Mattila and Melnikov on the existence of lime?0 ?|?-z|>e (? - z)-1dµ(?) for finite Borel measures µ in the plane.

In this note we present a simple proof of an interesting recent result of Mattila and Melnikov [5] on the existence, in the principal value sense, of the Cauchy transform of a finite Borel measure .To be precise, given a complex finite Borel measure M, in the plano, set CL(z) = lim (Cz) -1 dp(~), E-0 IC-zliE whenever the limit exists.The main result of [5] is that, given a rectifiable curve P, the limit in (1) exists almost everywhere with respect to are length on r.
Since any rectifiable curve is the union of a set of zero length and a countable family of Lipschitz graphs, in proving the above statement one can assume that the curve under consideration is a Lipschitz graph.Now, if ¡t is supported on P the result follows from the Coifman-Meyer-McIntosh [2] L2 estimates for the Cauchy integral en Lipschitz graphs .Thus, the point of the Mattila-Melnikov Theorem is that there is no restraint whatsoever on the support of F¿,.
The a.e.existente of the principal value (1) is a consequence, via standard argumenta, of a weak type estimate which we now proceed to describe in the setting of Ahlfors-David regular curves; the most general class of rectifiable curves for which the Caüchy.integral satisfies L 2 estimates [3] .
Let I' be a rectifiable curve and A be one dimensional Hausdorff measure restricted to 1'.One says that r is Ahlfors-David regular whenever A(A(z, r) fl r) < Cr, z E e, for some constant C and any dise 0(z, r) with center z and radius r.Set and In [5] the following inequality was proved for Lipschitz graphs .
for all complex fpite Borel measures M in tltie plane.
We proceed now to explain the main idea in our proof of (2) .The interested reader will find complete details in Section 2 .
The L2 inequalities for C* proved by David, combined with standard Calderón-Zygmund Theory on homogeneous spaces, tell us that (2) is true when p is supported on r .In particular, for measures p supported on I', one has for any compact subset K of 1'.
A{z : ¡Clc(z)j > t} < Ct-1jipli .Now, a weak type inequality for a linear operator has a dual formulation (see Section 2) which, in the case of (3) for measures supported on I' turns out to be On the other hand (2) (without any restriction on the support of p,) is equivalent to (3) (without any restriction on the support of IL), because a suitable Cotlar type inequality is available in our setting, and (3) (without any restriction on the support of M) can be dualized to for any compact subset K of I'.Therefore we are left with the task of showing that for b E L'(K), K compact in I', lb¡ <_ 1 and IC(bdA)1 <_ 1 A-a.e. imply IC(bdA)1 < C = C(F) on C\I', which turns out to be a simple matter [1, p. 110] .

Proof of the Theorem
We start by showing how to dualize a weak type inequality for a linear operator.
Let X and Y be locally compact Hausdorff spaces, A a positive Radon measure on X and T a bounded linear operator from the space M(X) of all complex finite :Radon measures on X into CO(Y), the space of continuous functions on Y vanishing at oo. Assume, furthermore, that the transpose of T, say T*, sends boundedly M(Y) into Co(X).
Lemma.The following statements are equivalent.
Moreover the least constants in (i) and (ii), say cl and c2, satisfy A-I < el e-I < A, for some absolute constant A .Proof. .For (i) =~> (ii) see [1,Th . 23,p . 107] .The reverse implication is very simple.It is clearly enough to prove that for each measure fi one has A{x : ReT*lc(x) > 1} < ClIMII .Take a compact subset K of X such that ReT*h > 1 on K and the left hand side of the last inequality is not greater than twice A(K) .By (ii), A(K) _< C f bdA, for some b E L'(K), 0 <_ b <_ 1 A-a .e., ITb1 <_ 1 on Y .Thus J bdA < J bReT*pdA = Re 1 Tbdp < JIp,II, which gives (i) .
To apply the lerrima it will be necessary to replace CEp  with a constant C independent of E .Applying the Lemma to X = 1', A = A, Y = r, T = KE, we conclude that given any compact K C I' we can find b E L'(K), 0 _< b _< 1 A-a.e ., IKEbI <_ 1 on r such that A(K) <_ C f bdA.It is now easy to ascertain as in [1, p. 110] that, for some constant C(1) depending only on f, IKE bI < C(1') on C.
A second appeal to the Lemma for X = P, A = A, Y = C, T = KE, gives (5) for all finite Borel measures p and some constant C independent of E.
To complete the proof we only need to apply a well known technology.The subspace of M(C) defined by S = {W (z) dzr+v}, where w E Có (C) and v is supported on a closed set of vanishing A-measure, is norm dense in M(C), and clearly Cp(z) exists A-a.e. for p E S. Since (5) holds with KE replaced by CE because of (4), C extends from S to an operator, which we still call C, defined on M(C) and satisfying (3) for each p E M(C) .Now one proves, exactly as in [4, p. 56] that for each p, 0 < p < 1, there exists a constant Cp with The history of dualizing a weak type inequality, as far as the author knows, is as follows.lt appears for the first time in a paper by N .X. Uy (Removable sets of analytic functions satisfying a Lipschitz condition, Ark. Mat. 17 (1.979),19-27), where it was used to characterize the renovable sets for Lipschitz analytic functions in the plano as those having zero area.S .Hruscev, who had also been working in come removability problems for analytic functions (The problem of simultaneous approximation a,nd rernoval of singularities of Cauchy-type integrals, Proceedings of the Steklov Inst . of Math 4 (1979), 133-203) found a neat proof of Uy's result which was reported in a paper by the author (C"à pproximation by solutions of elliptic equations, and Calderón-Zygmund operators, Duke Math.J. 55 (1987), 155-187) .Davie and Oksendal gave essentially the proof of the Lennna in Section 2 while proving that the Denjoy conjecture follows from Calderón L2 estimates for the Cauchy integral on Lipschitz graphs with srnall Lipschitz constant (see their joint paper Analytic capacity and differentiability properties of finely harmonic funetions, Acta Math .149 (1982), 127-152) .More recently Murai has used dual versions of weak type inequalities to relate analytic capacity and length on chord-are curves ("A real variable method for the Cauchy transform and analytic capacity", Lecture Notes in Math.1307, Springer-Verlag, New York, 1988) .
Finally, we would like to point out a couple of unsolved problems involving analytic capacity.
Let rn be a locally finito positive Borel rneasure in the plano.One would like to show that an inequality of the type , is equivalent to where y is analytic capacity.Thus one is load to ask whether or not an absolute constant A ca,n be found such that (6) C*ic(z) < CpMp(z)+Cr,M(ICpIP)1/P(z) .Kolmogorov's inequality [4, p. 5] and (6) now give (2) .

A
remarks ra{z : C* p(z) > t} :5 Ct, -r 11PI1 m(K) < Cy( .K), for all compact sets K, y{z : C * p(z) > t} < At-I for all finito Borel measures p in the plano.lf (7) were true, then nrany aspects of the classical potential theory of the Riesz kernels would have their counterpart for the complex potential l/z .