Tuesday, September 24th, 2013
11:00am - 12:00pm, in McCormack 1-617

Loredana Lanzani

University of Arkansas

Harmonic Analysis Techniques in Several Complex Variables

Abstract: In this talk I will show how the theory of singular integral operators, as first conceived by Calderon and Zygmund and subsequently developed by Cofman-McIntosh-Meyer, David, David-Semmes, can be employed to study operators that are not of Calderon-Zygmund type by way of a clever ``comparison argument'' with certain, Cauchy-type, CZ integrals. This comparison argument was first introduced by Kerzman and Stein to study the Szego projection for a $C^\infty$-smooth domain, and was later adapted by Ligocka to the case of the Bergman projection. The Szego and Bergman projections (that is, the orthogonal projections of the Lebesgue space $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions) are among the fundamental objects of complex function theory; like the Cauchy integral, they have analytic properties that are closely related to the regularity and geometry of the ambient domain. Unlike the Cauchy integral, they are not CZ operators (far from it): for instance, for these operators L^2 regularity (which they satisfy automatically) does not imply L^p regularity for $p\neq 2$. In this talk I will begin by reviewing the most relevant features of the classical Cauchy integral for a planar curve.I will then move on to the (surprisingly more involved) construction of the Cauchy integral for a hypersurface in $\mathbb C^n$. I will then present recent results joint with E. M. Stein concerning the regularity properties of this integral and their relation with the geometry of the hypersurface. I will conclude by presenting a new twist on the original ``comparison argument'' by Kerzman-Stein and Ligocka that yields L^p-regularity of the Szego and Bergman projections under what are the currently known minimal assumptions on the domain's boundary regularity.