Minimal $\mathcal{L}^{p}$-densities with prescribed marginals

We derive sharp lower bounds for $\mathcal{L}^{p}$-functions on the $n$-dimensional unit hypercube in terms of their $p$-ths marginal moments. Such bounds are the unique solutions of a system of constrained nonlinear integral equations depending on the marginals. For square-integrable functions, the bounds have an explicit expression in terms of the second marginals moments.