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 $\displaystyle \int_{V}k(x,\ y)f(y)dy$ defined on a homogeneous cone $V$. The results of this dissertation are then applied to an important class of operators such as RiemannLiouville'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.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 DOI:
 10.48550/arXiv.2206.08987
 arXiv:
 arXiv:2206.08987
 Bibcode:
 2022arXiv220608987V
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 4602;
 4302;
 4402