The L p boundedness of Riesz transforms associated with divergence form operators (original) (raw)

Let A be a divergence form elliptic operator associated with a quadratic form on Ω where Ω is the Euclidean space ℝ n or a domain of ℝ n . Assume that A generates an analytic semigroup e -t A on L 2 (Ω) which has heat kernel bounds of Poisson type, and that the generalized Riesz transform ∇A -1/2 is bounded on L 2 (Ω). We then prove that ∇A -1/2 is of weak type (1,1), hence bounded on L p (Ω) for 1<p≤2. No specific assumptions are made concerning the Hölder continuity of the coefficients or the smoothness of the boundary of Ω.