On Applications of FBI Transforms to Wave Front Sets

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Addis Ababa University


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



On Applications of FBI Transforms