Fractional integrals of generalized functions

Abstract

Riemann-Liouville integrals (of fractional order) of distributions can be obtained from the convolution theory of distributions whose support is bounded on the left. It is more difficult to define Weyl integrals of distributions; and multiplication by powers of the variable or integration of fractional order with respect to a power of the variable, both of which occur in applications, are not feasible in distribution theory. An alternative approach is based on the remark that the operator of Riemann-Liouville integrals and that of Weyl integrals are adjoint to each other. Constructing a space of testing functions on which one of these operators is continuous enables one to define the other for a corresponding class of generalized functions. Moreover, the testing function spaces can be constructed so that multiplication by a power of the variable and integration with respect to such a power are permissible. The resulting fractional integrals have applications to the Hankel transformation, to some singular differential operators, and to certain integral equations of the first kind.