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SUMMARY:Gennady Uraltsev (Cornell University)
DTSTART:20201209T170000Z
DTEND:20201209T180000Z
DTSTAMP:20260423T035707Z
UID:HAeS/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/HAeS/8/">Som
 e results in Banach space-valued time frequency analysis</a>\nby Gennady U
 raltsev (Cornell University) as part of Harmonic analysis e-seminars\n\n\n
 Abstract\nSIO (Singular Integral Operator) theory and\, Calderón-Zygmund 
 theory specifically\, developed starting from the '60s\, provides a vast a
 rray of tools for dealing with operators that resemble the Hilbert transfo
 rm\n$$\n\\mathrm{H}f(x):= \\int_{\\mathbb R}f(x-y)\\frac{d y}{y}\,\n$$\n\n
 an ubiquitous operator in Complex Analysis\, semi-linear PDEs\, and many o
 ther branches of mathematics. Results valid for -valued functions were ext
 ended to Banach spaces-valued functions thanks to Bourgain's groundbreakin
 g work on the deep relation between Banach space geometry and boundedness 
 properties of vector-valued SIOs.\n\nScalar-valued bounds for multilinear 
 SIOs\, like the bilinear Hilbert transform\n\n$$\n\\mathrm{BHT}[f_{1}\,f_{
 2}](x)=\\int_{\\mathbb R} f_{1}(x-t) f_{2}(x+t) \\frac{d t} {t}\,\n$$\n \n
 are classic in time-frequency-scale analysis. Banach-space valued results 
 have appeared only in the last couple of years. The well understood connec
 tions with Banach space geometry from linear theory are just starting to b
 e investigated.\n\nOpen questions and generalizations to non-commutative a
 nalysis abound and would come hand-in-hand with progress in understanding 
 SIOs with worse singularities than of Calderón-Zygmund type that can ofte
 n be realized as SIO-valued CZ operators.\n
LOCATION:https://researchseminars.org/talk/HAeS/8/
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