Applied Math And Analysis Seminar
Monday, March 21, 2016, 3:00pm, TBA
Loredana Lanzani (Syracus University)
Harmonic Analysis Techniques in Several Complex Variables
Abstract:- This talk concerns recent joint work with E. M. Stein on the extension to higher dimension of Calder\'ons and Coifman-McIntosh-Meyers seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset \mathbb{C}$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel 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 $H(w, z)$. 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 $C^{\infty}$-smooth, convex domain $D$: while these conditions on $D$ can be relaxed a bit, if the domain is less than $C^2$-smooth (much less Lipschitz!) Lerays construction becomes conceptually problematic.
In this talk I will present (a), the construction of the Cauchy-Leray kernel and (b), the (b) the $L^p(bD)$-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domains 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 Szeg\"o and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).
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