On Applications Of Fbi Transforms To Wave Front Sets

In this thesis, we study the application of FBI transforms to the C1; analytic andrnGevrey wave front sets of functions. We characterize the C1 wave front set of a functionrnby providing a simpler proof of a result by Berhanu and Hounie. To characterize thernanalytic wave front set, we generalize the work of Berhanu and Hounie [10] to twornpolynomials in the generating function of the FBI transform they de_ne. The Gevreyrnwave front set is characterized _rst as in the paper of Berhanu and Hounie and thenrngeneralized to two polynomials.rnFinally, we apply the standard FBI transform to study the microlocal smoothness ofrnC2 solutions u of the _rst-order nonlinear partial di_erential equationrnut = f(x; t; u; ux)rnwhere f(x; t; _0; _) is a complex-valued function which is C1 in all the variables (x; t; _0; _)rnand holomorphic in the variables (_0; _): If the solution u is C2; _ 2 Char(Lu) andrnrni _([Lu; L_u]) < 0; then we show that _ 2= WF(u): Here WF(u) denotes the C1 wavernfront set of u and Char(Lu) denotes the characteristic set of the linearized operator

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