Integrable Systems, Analysis and Probability Seminar
Thursday, October 2, 2003, 4:00pm, 120 Physics
Svetlana Roudenko (Duke University)
Level Set Operators vs. Oscillatory Integral Operators
Abstract:- Both oscillatory integral operators and level set operators
appear naturally in the study of properties of degenerate Fourier
integral operators (for example, generalized Radon transforms).
The properties of oscillatory integral operators have a longer history
and are better understood. On the other hand, level set operators,
while sharing many common characteristics with oscillatory integral
operators, seem easier to handle.
We study L^2 estimates on level set operators and compare them with
what is known about oscillatory integral operators. The cases we
consider include operators in one dimension with arbitrary smooth
phase functions, operators in 2 dimensions with "non-degenerate"
and certain cases of the "degenerate" phase functions.
In particular, we discuss the level set version of Melrose-Taylor
transform. This Radon-type transform comes from the scattering theory
and is associated with a singular canonical relation.
Operators in higher dimensions are considered as well.
The estimates are formulated in terms of the Newton polyhedra and
type conditions.
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