Integral transforms with H-function kernels on $\LLL_{ν,r}$-Spaces

Abstract

Integral transforms $$(\mbox{\boldmath$H$}f)(x)=\int^\infty_0H^{m,n}_{\thinspace p,q} \left[xt\left|\begin{array}{c}(a_i,α_i)_{1,p}\\[1mm](b_j,β_j)_{1,q} \end{array}\right.\right]f(t)dt$$ involving Fox&#39;s $H$-functions as kernels are studied in the spaces $\Ls_{ν,r}$ of functions $f$ such that $$\int^\infty_0|t^νf(t)|^r\frac{dt}t<\infty\quad(1\ \eqls\ r<\infty, \ ν\in\Rs).$$ Mapping properties such as the boundedness, the representation and the range of the transforms \boldmath$H$ are given.