ABSTRACT:

We present a condition on accretive matrix functions, called p-ellipticity, and discuss its applications to the Lp theory of elliptic PDE with complex coefficients. The examples we consider concern:

(1) generalized convexity of power functions (Bellman functions),
(2) dimension-free bilinear embeddings,
(3) Lp-contractivity of semigroups,
(4) holomorphic functional calculus,
(5) regularity theory of elliptic PDE with complex coefficients,
(6) maximal Lp regularity for divergence-form operators with Neumann
boundary conditions.

Example (5) is due to Dindoš and Pipher. The condition arises from studying uniform positivity of a quadratic form associated with the
matrix in question on one hand, and the Hessian of a power function $|z|^p$ on the other.

The talk is based on joint work with Andrea Carbonaro.