Integral transforms with H-function kernels on ${\text{LLL}}_{\nu ,r}$-Spaces

Integral transforms (\boldmath$H$f)(x)=\int^\infty_0H^{m,n}_{\thinspace p,q} \left[xt\left|\begin{array}{c}(a_i,\alpha_i)_{1,p}\\[1mm](b_j,\beta_j)_{1,q} \end{array}\right.\right]f(t)dt involving Fox’s $H$-functions as kernels are studied in the spaces ${\text{Ls}}_{\nu ,r}$ of functions $f$ such that $${\int }_0^{\mathrm{\infty }}\mathrm{\mid }{t}^{\nu }f(t){\mathrm{\mid }}^r\frac{dt}{t}<\mathrm{\infty }(1\text{eqls}r<\mathrm{\infty },\nu \in \text{Rs})\mathrm{.}$$ Mapping properties such as the boundedness, the representation and the range of the transforms \boldmath$H$ are given.