We introduce a condition on accretive matrix functions, called -ellipticity, and discuss its applications to the theory of elliptic PDEs with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) -contractivity of semigroups, and (iv) holomorphic functional calculus. Recent work by Dindos and Pipher established close ties between -ellipticity and (v) regularity theory of elliptic PDEs with complex coefficients. The -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.
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Andrea Carbonaro, Oliver Dragičević, Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients. J. Eur. Math. Soc. 22 (2020), no. 10, pp. 3175–3221DOI 10.4171/JEMS/984