A set of necessary and sufficient conditions for the Brascamp–Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank 1. This complements the result of Barthe concerning the case where the linear maps all have rank 1. Pushing our analysis further, we describe the case where the maps have either rank 1 or rank 2.

A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp–Lieb inequality to hold. We present an algorithm which generates such a list.

Let $F:(0, \infty) \times [0, \infty) \rightarrow \mathbb{R}$ be a function of Carathéodory type. We establish the inequality $$ \int_{\mathbb{R}^{N}} F( | x |, u(x) ) dx \leq \int_{\mathbb{R}^{N} } F( | x |, u^{\ast}(x)) dx. $$ where $u^{\ast}$ denotes the Schwarz symmetrization of $u$, under hypotheses on $F$ that seem quite natural when this inequality is used to obtain existence results in the context of elliptic partial differential equations. We also treat the case where $\mathbb{R}^N$ is replaced by a set of finite measure. The identity $$ \int_{\mathbb{R}^{N}} G(u(x)) dx = \int_{\mathbb{R}^{N}} G(u^{\ast}(x)) dx $$ is also discussed under the assumption that $G : [0,\infty) \rightarrow \mathbb{R}$ is a Borel function.

In this note we extend a theorem of Kwong and Zettl concerning the inequality

The Kwong-Zettl result holds for 1 ≤ p < ∞ and real numbers α, β, γ such that the conditions (i) β = (α + γ)/2, (ii) β > - 1 , and (iii) γ - 1 - p hold. Here the inequality is proved with β satisfying (i) for all α, γ except p — 1,-1 — p. In this case the inequality is false; however u is shown to satisfy the inequality

This paper is a continuation of [3] which initiated a systematic study of sufficient conditions for the weighted interpolation inequality of sum form, 1.1 to hold. Here ϕ, θ are non-negative functions of m, j, p, q, r, Ω is a bounded or unbounded domain in Rn, ∊ belongs to an interval Γ=(0, ∊0), u is in a certain Banach space E(Ω), and N, W, P are measurable real functions satisfying N≧ 0, W, P > 0, as well as additional conditions stated below. Finally the constant K does not depend on u although it may depend on the other parameters.