Pseudo-differential operators involving Watson transform

Abstract

A class of pseudo-differential operators (p.d.o.'s) associated with the general Fourier kernel studied by Hardy and Titchmarsh is defined. A symbol class T ^m is introduced. It is shown that the p.d.o.'s associated with the symbol are continuous linear mappings of the Braaksma and Schuitman space T(λ,μ) into itself. An integral representation of p.d.o. is obtained. Some special forms of the symbol are considered. It is shown that these p.d.o.'s and their products are bounded in certain Sobolev type space.

Keywords:

• Pseudo-differential operator
• Watson transform
• Sobolev type space
• Fourier transform

Keywords:

• 2000 Mathematics Subject Classifications:
• Primary 46F12
• 47G30