The FBI Transform and Microlocal Analysis in Ultradifferentiable Classes

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Addis Ababa University


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



FBI Transform, Microlocal Analysis, Ultradifferentiable Classes