On Hyper Singular Integral Operators over Weighted Sobolev Spaces

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.