Geometry
Seminar
Singular
Cauchy Integrals and
Hilbert
transforms on Curves and Surfaces
Professor Qian
Tao
University of Macau
Abstract
There
could be some confusion between Cauchy singular integrals and Hilbert transforms
on curves and surfaces. We justify the terminology Hilbert transforms by
defining it through Hardy spaces, following Bell and the others, and prove their Lp
boundedness, where the rage of p depends on the Lipschitz constant of
the curve or surface under study. Next we deduce he inner and outer
conjugate Poisson kernels, and thus the inner and outer Schwarz kernels on the
unit sphere, and the Fourier multiplier representations of the inner and outer
Hilbert transforms on the sphere. We finally prove the boundedness in all Lp, 1 < p < ¥, of the Hilbert
transforms on the sphere from the theory of the operator algebra of singular
integrals with monogenic Calderon-Zygmund type
kernels.
The
talk is based on a joint work between the speaker, A. Axelsson,
K-I. Kou and Y. Yan.
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Date:
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May 3, 2007 (Thursday)
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Time:
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3:00 – 4:00pm
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Place:
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Room 517, Meng Wah Complex, HKU
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