L 2 ESTIMATES ON CHORD - ARC CURVES 227 Theorem

We characterize those domains in the plane whose boundary is a chord arc curve in terms of some L 2 integrals, which are mainly a version of Green's theorem. As a consequence of this we obtain a \converse" to a theorem due to Laurentiev that states that for such domains harmonic measure and arc length are A 1 equivalent. Let ? be a locally rectiiable Jordan curve in the plane that passes through 1, and let + , ? be the two domains bounded by ?. Given a function f deened on ?, its Cauchy integral Cf(z) = Z ? f() ? z dd; z = 2 ? deenes an analytic function oo ?. denote their boundary values, then f (z) = 1 2 f(z) + 1 2i P.V. Z ? f() ? z dd; z 2 ?: G. David has shown in D] that the Cauchy integral is bounded in L 2 (?) if and only if ? is regular, that is, there exists a constant C such that for all z 0 2 C and all R > 0, the arclength of B(z 0 ; R) \ ? is at most CR, where B(z 0 ; R) denotes the ball centered at z 0 and radius R. Several proofs have been given of the boundedness of the Cauchy integral under stronger hypothesis on ?. We shall concentrate on the rst proof presented in C-J-S] which is based on complex variables methods. They show the result for Lipschitz graphs, i.e., ? = fx + iA(x) : x 2 Rg with A 0 2 L 1 : By following their argument very closely one can notice that the theorem is a consequence of the fact that for any F holomorphic in that decays to zero at 1, the following two integrals are equivalent: Z Z jF 0 (z)j 2 (z) dx dy = Z ? jFj 2 ds where (z) = dist(z; ?).