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SUMMARY:Loredana Lanzani (Syracuse University\, USA)
DTSTART:20211207T140000Z
DTEND:20211207T150000Z
DTSTAMP:20260422T201545Z
UID:CAvid/59
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/59/">T
 he Cauchy-Szegö projection and its commutator for domains in $\\mathbb C^
 n$ with minimal smoothness</a>\nby Loredana Lanzani (Syracuse University\,
  USA) as part of CAvid: Complex Analysis video seminar\n\nLecture held in 
 N/A.\n\nAbstract\nLet $D\\subset\\C^n$ be a bounded\, strongly pseudoconve
 x domain whose boundary $bD$ satisfies the minimal regularity condition of
  class $C^2$.  A 2017 result of Lanzani and Stein states that \nthe Cauchy
 -Szegö projection $S_\\omega$ defined with respect to any Leray Levi-like
  measure $\\omega$ is bounded in $L^p(bD\, \\omega)$ for any $1 < p < \\in
 fty$.\n(For this class of domains\, induced Lebesgue measure  is Leray Lev
 i-like.)\n  Here we show that $S_\\omega$\n  is in fact bounded in $L^p(bD
 \, \\Omega_p)$ for any $1 < p< \\infty$ and for any $\\Omega_p$ in the opt
 imal\n  class\n   of $A_p$ measures\, that is $\\Omega_p = \\psi_p\\sigma$
  where $\\sigma$ is induced Lebesgue measure and $\\psi_p$ is any Muckenho
 upt $A_p$-weight.\n   As an application\, we\n characterize boundedness an
 d compactness  in $L^p(bD\, \\Omega_p)$ for any $1 < p < \\infty$ and for 
 any $A_p$ measure $\\Omega_p$\, of the commutator $[b\, S_p]$ for any Lera
 y Levi-like measure $\\omega$. \n  We next introduce the notion of holomor
 phic Hardy spaces for $A_p$ measures\,\n   $1 < p < \\infty$\,  \n  and \n
  we characterize\n boundedness and compactness  in $L^2(bD\, \\Omega_2)$ o
 f the commutator \n $\\displaystyle{[b\,S_{\\Omega_2}]}$ of the Cauchy-Sze
 gö projection defined with respect to any \n $A_2$ measure $\\Omega_2$.\n
  Earlier closely related results \n  rely upon an asymptotic expansion\, a
 nd subsequent pointwise estimates\, of the Cauchy-Szegö kernel\, but thes
 e are unavailable in the settings of minimal regularity {of $bD$} and/or $
 A_p$ measures\; it turns out that the real harmonic analysis method of ext
 rapolation is an appropriate replacement for the missing tools.\n\n  \nThi
 s is joint work with Xuan Thinh Duong (Macquarie University)\, Ji Li (Macq
 uarie University) and Brett Wick (Washington University in St. Louis).\n
LOCATION:https://researchseminars.org/talk/CAvid/59/
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