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Finiteness of the Hölder-Brascamp-Lieb Constant Revisited

Philip T. Gressman

TL;DR

This paper provides a novel, graph-theoretic characterization of when the Hölder-Brascamp-Lieb constant is finite, complementing the classical BCCT criteria. It introduces graph decompositions and valid presentations, and establishes a factorization framework via normalized edge functions that converts the finiteness problem into a finite combinatorial construction. The main result shows finiteness is equivalent to the existence of a valid presentation, with a constructive proof that yields explicit bounds on the constant and a practical route to verification. The approach enables a finite, structured analysis of subspaces and paves the way for generalizations to mixed-norm inequalities and foliations, with potential connections to invariant polynomials and computational tractability.

Abstract

Abstract Hölder-Brascamp-Lieb inequalities have become a ubiquitous tool in Fourier analysis in recent years, due in large part to a theorem of Bennett, Carbery, Christ, and Tao (2008,2010) characterizing finiteness of the Hölder-Brascamp-Lieb constant. Here we provide a new characterization of a substantially different nature involving directed graphs of subspaces. Its practical value derives from its complementary nature to the Bennett, Carbery, Christ, and Tao conditions: it creates a means by which one can establish finiteness of the Hölder-Brascamp-Lieb constant by analysis of a well-chosen, finite list of subspaces rather than by checking conditions on all subspaces of the underlying vector space. The proof is elementary and is essentially an "explicitization" of the semi-explicit factorization algorithm of Carbery, Hänninen, and Valdimarsson (2023).

Finiteness of the Hölder-Brascamp-Lieb Constant Revisited

TL;DR

This paper provides a novel, graph-theoretic characterization of when the Hölder-Brascamp-Lieb constant is finite, complementing the classical BCCT criteria. It introduces graph decompositions and valid presentations, and establishes a factorization framework via normalized edge functions that converts the finiteness problem into a finite combinatorial construction. The main result shows finiteness is equivalent to the existence of a valid presentation, with a constructive proof that yields explicit bounds on the constant and a practical route to verification. The approach enables a finite, structured analysis of subspaces and paves the way for generalizations to mixed-norm inequalities and foliations, with potential connections to invariant polynomials and computational tractability.

Abstract

Abstract Hölder-Brascamp-Lieb inequalities have become a ubiquitous tool in Fourier analysis in recent years, due in large part to a theorem of Bennett, Carbery, Christ, and Tao (2008,2010) characterizing finiteness of the Hölder-Brascamp-Lieb constant. Here we provide a new characterization of a substantially different nature involving directed graphs of subspaces. Its practical value derives from its complementary nature to the Bennett, Carbery, Christ, and Tao conditions: it creates a means by which one can establish finiteness of the Hölder-Brascamp-Lieb constant by analysis of a well-chosen, finite list of subspaces rather than by checking conditions on all subspaces of the underlying vector space. The proof is elementary and is essentially an "explicitization" of the semi-explicit factorization algorithm of Carbery, Hänninen, and Valdimarsson (2023).

Paper Structure

This paper contains 7 sections, 5 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Valid presentations imply finiteness
  3. Properties of graph decompositions and balanced weights
  4. Factorizations
  5. Finiteness of the constant
  6. Existence of valid presentations
  7. Closing Comments

Key Result

Theorem 1

For given data $(H,\{\pi_i\}_{i=1}^N, \{\tau_i\}_{i=1}^N)$, the Hölder-Brascamp-Lieb inequality brascamplieb holds for some finite constant $C$ and all nonnegative measurable functions $\{f_i\}_{i=1}^N$ if and only if there exists $({\mathcal{G}},\{\theta_i\}_{i=1}^N)$ which is a valid presentation

Theorems & Definitions (9)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof