On the existence of some integral transforma as weak functions

Abstract

The definition of integral transforms of the type

as weak functions in the terminology of Lighthi11⁷ and Jones⁴ is obtained. With respect to the theory of Fourier integral operators, both the amplitude function h and the phase function ψ vary in the Hörmander classes², but the phase ψ, satisfies conditions less restrictive than the ones used by Traves¹². Some concrete examples of h are given such as a good function as in Jones⁴ so that the class of the amplitude functions h ends up to be wide. The interdependence between the parameter of the phase function ψ, and the positivity condition required is also studied. More specific phase functions are used than the ones by Trbves¹², in a way that our results can become more concrete. Some properties concerning the derivatives of the considered weak functions are also discussed.

Keywords:

• The definition of integral transforms as weak functions