Global gradient bounds for dissipative diffusion operators (original) (raw)

Let L be a second order elliptic operator on R d with a constant diffusion matrix and a dissipative (in a weak sense) drift b ∈ L p loc with some p > d. We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measure µ satisfying the equation L * µ = 0 and that the closure of L in L 1 (µ) generates a Markov semigroup {T t } t≥0 with the resolvent {G λ } λ>0 . We prove that, for any Lipschitzian function f ∈ L 1 (µ) and all t, λ > 0, the functions T t f and G λ f are Lipschitzian and sup