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In this paper, by establishing the $L^p$-$L^q$ estimate and Sobolev estimatesfor parabolic partial differential equations with a singular first order termand a Lipschitz first order term, a new Zvonkin-type transformation is givenfor stochastic differential equations with singular and Lipschitz drifts. Theassociated Krylov's estimate is established. As applications, Harnackinequalities are established for stochastic equations with H\"older continuousdiffusion coefficient and singular drift term without regularity assumption.

A Zvonkin's transformation for stochastic differential equations with singular drift and related applications