  We prove an $L^p(\Omega)\times L^q(\Omega)\times L^r(\Omega)\rightarrow
L^1(\Omega\times (0,\infty))$ 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 rather naturally expressed in terms of a
condition known as $p$-ellipticity. Furthermore, we give applications to
paraproducts and square functions associated with the corresponding operator
semigroups.
