Loredana Lanzani, Syracuse
Title:聽 Harmonic Analysis techniques in Several Complex Variables
Abstract:聽This talk concerns the application of relatively classical tools from real harmonic analysis (namely, the聽T(1)-theorem for spaces of homogenous type) to the novel context of several complex variables. Specifically, I will present recent joint work with E. M. Stein (Princeton U.) 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 $D\subset C$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel:聽$H(w, z)=\frac{1}{2\pi i}\frac{dw}{w-z}$ is that it is holomorphic (that is, analytic) as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogue of聽H(w, z). This is because of geometric obstructions (the Levi problem), which 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 $C^\infty$-smooth, convex domain聽D: while these conditions on聽D can be relaxed a bit, if the domain is less than聽C^2-smooth (never mind Lipschitz!) Leray's construction becomes conceptually problematic. In this talk I will present (a), the construction of the Cauchy-Leray kernel and (b), the聽L^p(bD)-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity -- in the case of a planar domain these are akin to Lipschitz boundary, but in our higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called ``T(1)-theorem technique'' from real harmonic analysis. Time permitting, I will describe applications of this work to complex function theory - specifically, to the Szego and Bergman projections (that is, the orthogonal projections of聽L^2 onto, respectively, the Hardy and Bergman spaces of holomorphic functions). References:
[C] Calderon A. P, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. 74 no. 4, (1977) 1324-1327.
[CMM] Coifman R., McIntosh A. and Meyer Y., L'integrale de Cauchy definit un operateur borne sur L^2 pour les courbes Lipschitziennes, Ann. of Math. 116 (1982) no. 2, 361-387.
[L] Lanzani, L. Harmonic Analysis Techniques in Several Complex Variables, Bruno Pini Mathematical Analysis Seminar 2014, 83-110, Univ. Bologna Alma Mater Studiorum, Bologna.
[LS-1] Lanzani L. and Stein E. M., The Szego projection for domains in C^n with minimal smoothness, Duke Math. J. 166 no. 1 (2017), 125-176.
[LS-2] Lanzani L. and Stein E. M., The Cauchy Integral in C^n for domains with minimal smoothness, Adv. Math. 264 (2014) 776-830.
[LS-3] Lanzani L. and Stein E. M., The Cauchy-Leray Integral: counter-examples to the L^p-theory, Indiana Math. J., to appear.