Bilinear embedding for divergence-form operators with negative potentials

Let $\Omega \subseteq \mathbb{R}^d$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^\infty$ coefficients, and $V$ a locally integrable function on $\Omega$ whose negative part is subcritical. We consider the operator $\mathscr{L} = -\mathrm{div}(A\nabla) + V$ with mixed boundary conditions on $\Omega$. We extend the bilinear inequality of Carbonaro and Dragi\v{c}evi\'c [15], originally established for nonnegative potentials, by introducing a novel condition on the coefficients that reduces to standard $p$-ellipticity when $V$ is nonnegative. As a consequence, we show that the solution to the parabolic problem $u'(t) + \mathscr{L} u(t) = f(t)$ with $u(0)=0$ has maximal regularity on $L^p(\Omega)$, in the same spirit as [13]. Moreover, we study mapping properties of the semigroup generated by $-\mathscr{L}$ under this new condition, thereby extending classical results for the Schr\"{o}dinger operator $-\Delta + V$ on $\mathbb{R}^d$ [8,47].