  We prove a dimension-free $L^p(\Omega)\times L^q(\Omega)\times
L^r(\Omega)\rightarrow L^1(\Omega\times (0,\infty))$ embedding for triples of
elliptic operators in divergence form with complex coefficients and subject to
mixed boundary conditions on $\Omega$, and for triples of exponents
$p,q,r\in(1,\infty)$ mutually related by the identity $1/p+1/q+1/r=1$. Here
$\Omega$ is allowed to be an arbitrary open subset of $\mathbb{R}^d$. 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.
