Lp Uniqueness of Non-symmetric Diffusion Operators with Singular Drift Coefficients: I. The Finite-Dimensional Case

Two uniqueness results for C0 semigroups on weighted Lp spaces over Rn generated by operators of type Δ+β·∇ with singular drift β are proven. A key ingredient in the proofs is the verification of some kind of “weak Kato inequality” which seems to break down exactly for those drift singularities where Lp uniqueness breaks down as well.