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Using Zvonkin's transform and the Poisson equation in $R^d$ with a parameter,we prove the averaging principle for stochastic differential equations withtime-dependent H\"older continuous coefficients. Sharp convergence rates withorder $(\alpha\wedge1)/2$ in the strong sense and $(\alpha/2)\wedge1$ in theweak sense are obtained, considerably extending the existing results in theliterature. Moreover, we prove that the convergence of the multi-scale systemto the effective equation depends only on the regularity of the coefficients ofthe equation for the slow variable, and does not depend on the regularity ofthe coefficients of the equation for the fast component.

Strong and weak convergence in the averaging principle for SDEs with H\"older coefficients