Norm Inequalities for Integral Operators on Cones

TL;DR

This paper investigates the boundedness of integral operators on weighted spaces over cones in Euclidean space, applying results to fractional and Laplace operators, and generalizing Hardy's inequality.

Contribution

It provides new bounds for integral operators on cones and extends classical inequalities to higher-dimensional cone domains.

Findings

Established $[L^{p}, L^{q}]$-boundedness for integral operators on cones.

Applied results to fractional and Laplace operators.

Generalized Hardy's inequality to ${f R}^n$ domains of positivity.

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 $\displaystyle \int_{V}k(x,\ y)f(y)dy$ 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.