  In this paper we formulate new curvature functions on $\mathbb{S}^n$ via
integral operators. For certain even orders, these curvature functions are
equivalent to the classic curvature functions defined via differential
operators, but not for all even orders. Existence result for antipodally
symmetric prescribed curvature functions on $\mathbb{S}^n$ is obtained. As a
corollary, the existence of a conformal metric for an antipodally symmetric
prescribed $Q-$curvature functions on $\mathbb{S}^3$ is proved. Curvature
function on general compact manifold as well as the conformal covariance
property for the corresponding integral operator are also addressed, and a
general Yamabe type problem is proposed.
