A fundamental fact on Fourier analysis is that {\mathcal F} is a unitary
transform from $L^2({\mathbb R}^n)$ on the
"time" space to itself on the "frequency(=phase)"
space. A important class of singular integral operators is defined
via the Fourier transform. For example the Hilbert transform is a typical example of sigular integral operators realized
by the Fourier multiplier. A more advanced example will be a class of
PSIDO.
In this talk I will present a decomposition formula of the "frequency"
space. This talk shall be oriented to appications
of this new decomposition formula. If time permits, I will allude to technical
details. Finally, the material of this talk is
the following paper by the speaker, which will be published from Michigan
Mathematical Journal.
Seminar on Analysis at University of Tsukuba