Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

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.