  We study lower bounds on multilinear operators with Gaussian kernels acting
on Lebesgue spaces, with exponents below one. We put forward natural conditions
when the optimal constant can be computed by inspecting centered Gaussian
functions only, and we give necessary and sufficient conditions for this
constant to be positive. Our work provides a counterpart to Lieb's results on
maximizers of multilinear operators with real Gaussian kernels, also known as
the multidimensional Brascamp-Lieb inequality. It unifies and extends several
inverse inequalities.
