Settimana 19/06/2022 - 23/06/2022

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Università degli Studi Roma Tor Vergata
Dipartimento di Matematica

 

DIFFERENTIAL EQUATIONS SEMINAR

Date: 20 Jun 2023

Schedule: 14:30 - Rome Time

Where: Conference Room "R. Dal Passo"

Title: " Applications of the Calderòn-Zygmund theory to holomorphic singular integrals "

Speaker: Loredana Lanzani (Università di Bologna)

Abstract: Let D be a domain in complex Euclidean space Cn with the complex dimen- sion n being at least 2, so that the tool of conformal mapping is unavailable. The classical methods for the Cauchy-Szegő projection, SD , associated to such D rely on apriori esti- mates of the Cauchy-Szegő kernel and are most effective for smooth D (that is, D of class C∞). In 1978 Norberto Kerzman and Eli Stein pioneered a new approach based on an operator identity that relates SD to a holomorphic, Cauchy-type singular integral operator whose kernel had been constructed in 1969 independently by G. Khenkin and E. Ramirez de Arellano. The Kerzman-Stein identity is the earliest evidence pointing to a deep connec- tion between several complex variables and the Calderón-Zygmund theory. (Incidentally, Calderòn’s iconic result on the Cauchy transform for a Lipschitz planar curve also appeared in 1978.) As the Calderòn-Zygmund theory was further developed – notably, through the T (1) theorem – its connection with several complex variables also evolved and ultimately led to the resolution, in 2017, of the Lp-regularity problem for SD associated to strongly pseudoconvex D with minimal boundary regularity (namely, the class C2), along with the resolution of the analogous problem for the Bergman projection BD . In this talk I will highlight my collaboration with Stein on these problems. I will also present new developments, by several authors, that have stemmed from this work, specifically: the weighted Lp-regularity of SD and BD , and characterizations of (unweighted and weighted) Lp-regularity and Lp-compactness of the commutators [b, SD ] and [b, BD ] for strongly pseudoconvex D with minimal smoothness. I will close with open problems.

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)
Nota: Questo seminario fa parte delle attività del progetto "Dipartimento d'eccellenza" MatMod@TOV (2023-2027)