  This paper gives a concept of an integral operator defined on a manifold $M$
consisting of triple of points in $\mathbb{R}^{d}$ making up a regular
$3$-simplex with the origin. The boundedness of such operator is investigated.
The boundedness region contains more than the Banach range - a fact that
mirrors the spherical $L^{p}$-improving estimate. The purpose of this paper is
two-fold: one is to investigate into an integral operator over a manifold
created from high-dimensional regular simplices, two is to start a maximal
operator theory for such integral operator.
