On Hyper Singular Integral Operators over Weighted Sobolev Spaces

TL;DR

This paper investigates hyper singular integral operators on weighted Sobolev spaces over unbounded domains, focusing on controlling their singularities using weighted norms and analyzing the $ ext{ extpi}$-operator.

Contribution

It introduces new methods to control the singularity of hyper integral operators on weighted Sobolev spaces, especially for the $ ext{ extpi}$-operator, over unbounded domains.

Findings

Control of singularities via weighted Sobolev norms

Extension of hyper integral operator analysis to unbounded domains

Specific results for the $ ext{ extpi}$-operator

Abstract

In this paper we study singular integral operators which are hyper or weak over Lipschitz or Holder spaces and over weghted Sobolev spaces defined on unbounded domains in the standard $n$-D space $R^n$ for $n>0$. The $\pi$-operator in this case is one of the hyper integral operators which has been studied extensively than other hyper singular integral operators. It will be shown the control of singularity of such integral operators that are defined interms of Cauchy generating kernels by working on weghted Sobolev spaces $W^{p,k}(\Omega,|x|^{\zeta+epsilon}dx)$ for some $\epsilon>0$ and $\zeta $ some positive integer.