Communications of the
Korean Mathematical Society CKMS

In this article, we discuss classes of generalized functions for certain modified integral operator of Bessel-type involving Fox's $H$-function kernel. We employ a known differentiation formula of Fox's $H$-function to obtain the definition and properties of the distributional modified Bessel-type integral. Further, we derive a smoothness theorem for its kernel in a complete countably multi-normed space. On the other hand, using an appropriate class of convolution products, we derive axioms and establish spaces of modified Boehmians which are generalized distributions. On the defined spaces, we introduce addition, convolution, differentiation and scalar multiplication and further properties of the extended integral.

Keywords: Fox's $H$-function, Bessel-type integral, Boehmians, generalized functions, distribution

MSC numbers: 46F12, 46T30