Minimal L^p-Densities with Prescribed Marginals

TL;DR

This paper establishes sharp lower bounds for L^p-functions on the hypercube based on their marginal moments, providing explicit solutions for square-integrable functions and solving related nonlinear integral equations.

Contribution

It introduces a novel method to derive unique lower bounds for L^p-functions with prescribed marginals, including explicit formulas for square-integrable cases.

Findings

Derived sharp lower bounds for L^p-functions based on marginals

Solved a system of nonlinear integral equations for these bounds

Provided explicit expressions for square-integrable functions

Abstract

We derive sharp lower bounds for L^p-functions on the n-dimensional unit hypercube in terms of their p-th 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.