Trilinear embedding for divergence-form operators with complex coefficients

We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.