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UID:20260608T121807EDT-9924ETH3Ij@132.216.98.100
DTSTAMP:20260608T161807Z
DESCRIPTION:Title:  Harmonic Analysis techniques in Several Complex Variabl
 es\n	Abstract: This talk concerns the application of relatively classical t
 ools from real harmonic analysis (namely\, the T(1)-theorem for spaces of 
 homogenous type) to the novel context of several complex variables. Specif
 ically\, I will present recent joint work with E. M. Stein (Princeton U.) 
 on the extension to higher dimension of Calderon's and Coifman-McIntosh-Me
 yer's seminal results about the Cauchy integral for a Lipschitz planar cur
 ve (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-dim
 ensional Cauchy kernel: $H(w\, z)=\frac{1}{2\pi i}\frac{dw}{w-z}$ 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 n
 o obvious holomorphic analogue of H(w\, z). This is because of geometric o
 bstructions (the Levi problem)\, which in dimension 1 are irrelevant. A go
 od candidate kernel for the higher dimensional setting was first identifie
 d by Jean Leray in the context of a $C^\infty$-smooth\, convex domain D: w
 hile these conditions on D can be relaxed a bit\, if the domain is less th
 an C^2-smooth (never mind Lipschitz!) Leray's construction becomes concept
 ually problematic. In this talk I will present (a)\, the construction of t
 he Cauchy-Leray kernel and (b)\, the L^p(bD)-boundedness of the induced si
 ngular integral operator under the weakest currently known assumptions on 
 the domain's regularity -- in the case of a planar domain these are akin t
 o Lipschitz boundary\, but in our higher-dimensional context the assumptio
 ns 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 Ber
 gman projections (that is\, the orthogonal projections of L^2 onto\, respe
 ctively\, the Hardy and Bergman spaces of holomorphic functions). Referenc
 es:\n	[C] Calderon A. P\, Cauchy integrals on Lipschitz curves and related 
 operators\, Proc. Nat. Acad. Sci. 74 no. 4\, (1977) 1324-1327.\n	[CMM] Coif
 man R.\, McIntosh A. and Meyer Y.\, L'integrale de Cauchy definit un opera
 teur borne sur L^2 pour les courbes Lipschitziennes\, Ann. of Math. 116 (1
 982) no. 2\, 361-387.\n	[L] Lanzani\, L. Harmonic Analysis Techniques in Se
 veral Complex Variables\, Bruno Pini Mathematical Analysis Seminar 2014\, 
 83-110\, Univ. Bologna Alma Mater Studiorum\, Bologna.\n	[LS-1] Lanzani L. 
 and Stein E. M.\, The Szego projection for domains in C^n with minimal smo
 othness\, Duke Math. J. 166 no. 1 (2017)\, 125-176.\n	[LS-2] Lanzani L. and
  Stein E. M.\, The Cauchy Integral in C^n for domains with minimal smoothn
 ess\, Adv. Math. 264 (2014) 776-830.\n	[LS-3] Lanzani L. and Stein E. M.\, 
 The Cauchy-Leray Integral: counter-examples to the L^p-theory\, Indiana Ma
 th. J.\, to appear.\n	\n	\n
DTSTART:20180216T183000Z
DTEND:20180216T193000Z
LOCATION:Room 920\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue 
 Sherbrooke Ouest
SUMMARY:Loredana Lanzani\, Syracuse
URL:https://www.mcgill.ca/mathstat/channels/event/loredana-lanzani-syracuse
 -285073
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