The Fbi Transform And Microlocal Analysis In Ultradifferentiable Classes

The FBI transform is a nonlinear Fourier transform that characterizes thernlocal/ microlocal smoothness and analyticity of functions (or distributions) inrnterms of appropriate decays. This characterization is very useful in studying thernlocal and microlocal regularity of solutions of partial differential equations.rnThe ultradifferentiable classes play an important role in the theory of differential equations as they provide an intermediate scale of spaces between Crn∞ andrnreal analytic functions.rnIn this thesis, we establish the boundedness of a class of FBI transforms inrnSobolev spaces. We characterize the ultradifferentiable wave front set by a classrnof FBI transforms. We also provide an application that shows how powerful arernthese generalized class of FBI transforms by exhibiting a result on microlocalrnregularity for solutions of first order nonlinear partial differential equations inrnthese classes, which can not be solved by the classical FBI transforms. Finally,rnwe use the FBI transform to characterize microlocal smoothness and microlocalrnultradifferentiablity on maximally real submanifolds

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