The FBI Transform and Microlocal Analysis in Ultradifferentiable Classes
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Date
2020-01-10
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Addis Ababa University
Abstract
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
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Keywords
FBI Transform, Microlocal Analysis, Ultradifferentiable Classes