Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. Informally the Schrödinger-type operators we consider are of the form L + μ · ∇ + ν where L is uniformly elliptic, μ is a vector-valued signed measure belonging to Kd,1 and ν is a signed measure belonging to Kd,2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains. AMS 2000 Mathematics Subject Classification: Primary: 58C60, 60J45; Secondary: 35P15, The research of this author is supported in part by a joint US-Croatia grant INT 0302167.