L2-boundedness of the Cauchy Integral Operator for Continuous Measures

1. Introduction. Let µ be a continuous (i.e., without atoms) positive Radon measure on the complex plane. The truncated Cauchy integral of a compactly supported function f in L p (µ), 1 ≤ p ≤ +∞, is defined by Ꮿ ε f (z) = |ξ −z|>ε f (ξ) ξ − z dµ(ξ), z ∈ C, ε > 0. In this paper, we consider the problem of describing in geometric terms those measures µ for which |Ꮿ ε f | 2 dµ ≤ C |f | 2 dµ, (1) for all (compactly supported) functions f ∈ L 2 (µ) and some constant C independent of ε > 0. If (1) holds, then we say, following David and Semmes [DS2, pp. 7–8], that the Cauchy integral is bounded on L 2 (µ). A special instance to which classical methods apply occurs when µ satisfies the doubling condition µ(2) ≤ Cµ((), for all discs centered at some point of spt(µ), where 2 is the disc concentric with of double radius. In this case, standard Calderón-Zygmund theory shows that (1) is equivalent to Ꮿ * f 2 dµ ≤ C |f | 2 dµ, (2) where Ꮿ * f (z) = sup ε>0 |Ꮿ ε f (z)|. If, moreover, one can find a dense subset of L 2 (µ) for which Ꮿf (z) = lim ε→0 Ꮿ ε f (z) (3) 269 270 XAVIER TOLSA exists a.e. (µ) (i.e., almost everywhere with respect to µ), then (2) implies the a.e. (µ) existence of (3), for any f ∈ L 2 (µ), and |Ꮿf | 2 dµ ≤ C |f | 2 dµ, for any function f ∈ L 2 (µ) and some constant C. For a general µ, we do not know if the limit in (3) exists for f ∈ L 2 (µ) and almost all (µ) z ∈ C. This is why we emphasize the role of the truncated operators Ꮿ ε. Proving (1) for particular choices of µ has been a relevant theme in classical analysis in the last thirty years. Calderón's paper [Ca] is devoted to the proof of (1) when µ is the arc length on a Lipschitz graph with small Lipschitz constant. The result for a general Lipschitz graph was obtained by Coifman, McIntosh, and Meyer in 1982 in the celebrated paper [CMM]. The rectifiable curves , for which (1) holds for the arc length measure µ on …