On Applications of FBI Transforms to Wave Front Sets
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Date
2016-06-30
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Addis Ababa University
Abstract
In this thesis, we study the application of FBI transforms to the C1; analytic and
Gevrey wave front sets of functions. We characterize the C1 wave front set of a function
by providing a simpler proof of a result by Berhanu and Hounie. To characterize the
analytic wave front set, we generalize the work of Berhanu and Hounie [10] to two
polynomials in the generating function of the FBI transform they de_ne. The Gevrey
wave front set is characterized _rst as in the paper of Berhanu and Hounie and then
generalized to two polynomials.
Finally, we apply the standard FBI transform to study the microlocal smoothness of
C2 solutions u of the _rst-order nonlinear partial di_erential equation
ut = f(x; t; u; ux)
where f(x; t; _0; _) is a complex-valued function which is C1 in all the variables (x; t; _0; _)
and holomorphic in the variables (_0; _): If the solution u is C2; _ 2 Char(Lu) and
i _([Lu; L_u]) < 0; then we show that _ 2= WF(u): Here WF(u) denotes the C1 wave
front set of u and Char(Lu) denotes the characteristic set of the linearized operator
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On Applications of FBI Transforms