  We consider a Poisson equation in $\mathbb R^d$ for the elliptic operator
corresponding to an ergodic diffusion process. Optimal regularity and
smoothness with respect to the parameter are obtained under mild conditions on
the coefficients. The result is then applied to establish a general diffusion
approximation for fully coupled multi-time-scales stochastic differential
equations with only H\"older continuous coefficients. Four different averaged
equations as well as rates of convergence are obtained. Moreover, the
convergence is shown to rely only on the regularities of the coefficients with
respect to the slow variable, and does not depend on their regularities with
respect to the fast component.
