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We obtain sharp weighted L p estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 < r < ∞ the norm of a sublinear operator on L r (w) is bounded by a function of the Ar characteristic constant of the weight w, then for p > r it is bounded on L p (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on L p (v) by the same increasing function of the r−1 p−1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.

We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1

Abstract. By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold (M, µϕ) having the Bakry-Emery curvature bounded from below. The embedding, acting on the Cartesian product of L p (M, µϕ) and L q (T * M, µϕ),

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