Norm Inequalities for Integral Operators on Cones

Abstract

In this dissertation we explore the $[L^{\mathrm{p}},L^q]$-boundedness of certain integral operators on weighted spaces on cones in ${\mathbb{R}}^n.$ These integral operators are of the type ${\int }_{V}k(x,y)f(y)dy$ defined on a homogeneous cone $V$. The results of this dissertation are then applied to an important class of operators such as Riemann-Liouville's fractional integral operators, Weyl's fractional integral operators and Laplace's operators. As special cases of the above, we obtain an ${\mathbb{R}}^n$ -generalization of the celebrated Hardy's inequality on domains of positivity. We also prove dual results.