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Quiver Brascamp-Lieb inequalities

Nicholas Hu

TL;DR

This work generalizes Brascamp–Lieb inequalities to bipartite quivers, establishing exact, necessary and sufficient criteria for finiteness of the quiver-extended constants via scaling and dimension inequalities. It refines Gaussian extremizer theory in the quiver setting, defining the Gaussian-optimized constant $\mathrm{BLCD}_{\mathrm{G}}(\mathcal{Q},\mathbf{p})$ and showing that Gaussian saturations can fail for general quiver data. A key contribution is proving the equivalence between arrow-based and subspace quiver formulations (Theorem T:equiv) and highlighting that, unlike the classical case, Gaussians need not attain the optimal constant in general (with explicit counterexamples). The sufficiency/necessity proofs rely on decomposing quivers into subspace components and Bennett–Carbery–Christ–Tao-type scaling arguments, clarifying when quiver inequalities mirror their Gaussian counterparts and when they do not. Together, these results sharpen our understanding of extremizers and constants in quiver Brascamp–Lieb inequalities and point to new directions for structure theory and potential applications.

Abstract

We formulate generalized Brascamp-Lieb inequalities for representations of bipartite quivers and establish necessary and sufficient conditions for such inequalities. Notably, we show contra Lieb that Gaussians do not saturate certain types of quiver Brascamp-Lieb inequalities.

Quiver Brascamp-Lieb inequalities

TL;DR

This work generalizes Brascamp–Lieb inequalities to bipartite quivers, establishing exact, necessary and sufficient criteria for finiteness of the quiver-extended constants via scaling and dimension inequalities. It refines Gaussian extremizer theory in the quiver setting, defining the Gaussian-optimized constant and showing that Gaussian saturations can fail for general quiver data. A key contribution is proving the equivalence between arrow-based and subspace quiver formulations (Theorem T:equiv) and highlighting that, unlike the classical case, Gaussians need not attain the optimal constant in general (with explicit counterexamples). The sufficiency/necessity proofs rely on decomposing quivers into subspace components and Bennett–Carbery–Christ–Tao-type scaling arguments, clarifying when quiver inequalities mirror their Gaussian counterparts and when they do not. Together, these results sharpen our understanding of extremizers and constants in quiver Brascamp–Lieb inequalities and point to new directions for structure theory and potential applications.

Abstract

We formulate generalized Brascamp-Lieb inequalities for representations of bipartite quivers and establish necessary and sufficient conditions for such inequalities. Notably, we show contra Lieb that Gaussians do not saturate certain types of quiver Brascamp-Lieb inequalities.
Paper Structure (7 sections, 7 theorems, 39 equations, 3 figures)

This paper contains 7 sections, 7 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Brascamp--Lieb inequalities
  3. Quiver Brascamp--Lieb inequalities
  4. Notation
  5. Conditions for quiver Brascamp--Lieb inequalities
  6. Examples of quiver Brascamp--Lieb inequalities
  7. Concluding remarks and questions

Key Result

Theorem 1.2

Let $\mathrm{BL}(\mathbf{B}, \mathbf{p})$ denote the smallest constant for which inequality bl holds, where $(\mathbf{B}, \mathbf{p}) \vcentcolon= ((B_1, \dots, B_m), (p_1, \dots, p_m))$; it is known as the Brascamp--Lieb constant for the Brascamp--Lieb datum$(\mathbf{B}, \mathbf{p})$. Then $\mathrm and the dimension condition hold. (Here $V \leq H$ means that $V$ is a subspace of $H$.)

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3: A bipartite quiver $\mathcal{Q}$ and its constituent subspace quivers $\mathcal{Q}_1$ and $\mathcal{Q}_2$.

Theorems & Definitions (20)

  • Definition 1.1: Brascamp--Lieb inequality
  • Theorem 1.2
  • Theorem 1.3
  • Example 1.4: Young's convolution inequality
  • Definition 1.5: Quiver
  • Definition 1.6: Quiver representation
  • Theorem 1.7
  • Definition 1.8: Quiver Brascamp--Lieb inequalities
  • Example 1.9
  • Theorem 1.10
  • ...and 10 more