Product and Commutator of pseudo-differential operators involving the coupled fractional Fourier transform

Abstract

The symbol class Λ(R × R × R × R) is discussed and it is shown
that the product of any two symbols from this class is again in Λ. The Pseudodifferential operators (p.d.o.) A(x, y,Dx
0
,y
) and A (x, y,Dx
0
,y
) involving the coupled
fractional Fourier transform Fα1,α2
associated with symbol classes are defined.
Product and Commutator for the p.d.o. and their boundedness results are obtained