Convexity of Power Functions and Bilinear Embedding for Divergence-form Operators with Complex Coefficients

Let LA, LB be the operators in divergence form associated with complex uniformly accretive matrix functions A,B on R. Take p > 1 and assume that for a.e. x ∈ R we have Re 〈A(x)ξ, Jpξ〉 > Cp|ξ| 2 for some Cp > 0 and all ξ ∈ C , where Jp(α + iβ) = α/p + iβ/p ′ for 1/p + 1/p′ = 1, and the same for B. Under this condition we prove that (f, g) 7→ 〈 ∇xe Af,∇xe Bg 〉 extends to a continuous bilinear embedding L(R) × L ′ (R) → L(R + ) with constants that are explicit and independent of the dimension n. Previous results of this kind covered only the case where A,B were real and equal. The proof rests on the analysis of the heat flow associated with a particular Bellman function, in the course of which we obtain the condition involving Jp. It arises from studying generalized convexity of power functions, i.e., from the (uniform) positivity of a quadratic form associated with A and Hess(|ζ|). By means of this condition we also characterize the contractivity on L of the associated semigroups in the case when the antisymmetric part of the imaginary part of A is constant. This extends previous results by Cialdea and Maz’ya. The proof is relatively short and does not involve full Bellman functions. Our condition is sharp in the sense that when it is not fulfilled, the (dimension-free) embedding in general fails, as does the L contractivity of the semigroup.