JournalsjemsVol. 22, No. 10pp. 3175–3221

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

  • Andrea Carbonaro

    Università degli Studi di Genova, Italy
  • Oliver Dragičević

    University of Ljubljana, Slovenia
Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients cover

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Abstract

We introduce a condition on accretive matrix functions, called pp-ellipticity, and discuss its applications to the LpL^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) LpL^p-contractivity of semigroups, and (iv) holomorphic functional calculus. Recent work by Dindos and Pipher established close ties between pp-ellipticity and (v) regularity theory of elliptic PDEs with complex coefficients. The pp-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.

Cite this article

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. 3175–3221

DOI 10.4171/JEMS/984