We introduce a condition on accretive matrix functions, called p -ellipticity, and discuss its applications to the L^p theory of elliptic PDEs with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) L^p -contractivity of semigroups, and (iv) holomorphic functional calculus. Recent work by Dindos and Pipher established close ties between p -ellipticity and (v) regularity theory of elliptic PDEs with complex coefficients. The p -ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on the one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz’ya.

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

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.

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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