Abstract. This talk concerns recent joint work with E. M. Stein on the extension to higher dimension of Calderon’s and Coifman-McIntosh-Meyer’s seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain D ⇢ C). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel: H(w,z)= (1/2\pi i)dw/(w-z),
is that it is holomorphic (that is, analytic) in D as a function of z. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogue of H(w, z). This is because of geometric obstructions (the Levi problem) that in dimension 1 are irrelevant.
A good candidate kernel for the higher dimensional setting was first identified by Jean Leray in the context of a C^1-smooth, convex domain: while the assumptions on the domain can be relaxed a bit, if the domain is less than C^2-smooth (much less Lipschitz!) Leray’s construction becomes conceptually problematic.
In this talk I will present (a), the construction of the Cauchy-Leray kernel and (b), the L^p(bD)-regularity of the induced singular integral operator under the weakest cur- rently known assumptions on the domain’s regularity – in the case of a planar domain these are akin to Lipschitz boundary. Time permitting, I will describe applications of this work to complex function theory (specifically, to the Szeg ̋o and Bergman projections).