Caltech/UCLA Joint Analysis Seminar

UCLA - MS 7608

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.