  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.
