Abstract: We propose algebraic criteria that yield sharp Holder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the Brascamp-Lieb inequality. Moreover, we will see that this result generalizes, Holder's inequality, Nelson's hypercontractivity as well as its reverse, and the sharp Young and revere sharp Young inequalities. Moreover we give two more applications: Prekopa-Leindler and Barthe inequality."

Abstract:

Let $\mathbb{N}$ be the set of all nonnegative integers. A set $A \subset \mathbb{N}$ is called a basis of $\mathbb{N}$ if every sufficiently large integer is a sum of $h$ elements from $A$, for some $h$. The smallest such $h$ is called the order of $A$. For example, the squares form a basis of order 4 and the primes form a basis of order 3 of $\mathbb{N}$. Erdos and Graham asked the following questions. If $A$ is a basis of $\mathbb{N}$ and $a \in A$, when is $A \setminus \{ a \}$ still a basis? It turns out that this is the case for all $a \in A$ except a finite number of exceptions. If $A \setminus \{ a \}$ is still a basis, what can we say about its order? These questions and related questions have been extensively studied. In this talk, we address these questions in the more general setting of an abelian group in place of $\mathbb{N}$. This is joint work with Victor Lambert and Alain Plagne.