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A Fubini Theorem for Grothendieck Functional Integrals

Haoran He, Qichen He

TL;DR

This work introduces Grothendieck functional integrals as a universal, measure-free framework for bounded linear functionals on projective tensor products, controlled by Grothendieck's constant $K_G$ and equivalent to bilinear forms via a Hilbert-space representation. It establishes a Hilbert-space factorization and an abstract Fubini theorem, then extends the theory to multilinear settings with a multilinear Fubini theorem. The authors provide concrete realizations in $L^p$ and $C(S)$ spaces, and demonstrate applications to pseudodifferential-type operators, including kernel representations, norm bounds, and spectral properties governed by $K_G$. The framework offers a unifying, dimension-free approach to bilinear and multilinear analysis with potential impact on operator theory, PDEs, and numerical methods. Overall, the paper presents a universal, structurally rich method for analyzing integral-type functionals via a fixed constant $K_G$, enabling efficient exchange of integration order and explicit operator representations across diverse function spaces.

Abstract

This paper systematically studies the subset of continuous linear functionals on the projective tensor product of Banach spaces whose norms are bounded by Grothendieck's constant $K_G$. We term such functionals Grothendieck functional integrals. The integral is defined as a linear functional on the projective tensor product space that satisfies the boundedness condition $|μ(x)| \leq K_G \|x\|_π$, where $K_G$ denotes Grothendieck's constant. We prove that such integrals admit a Hilbert space representation theorem and establish the corresponding abstract Fubini theorem to demonstrate that the order of integration may be interchanged. Furthermore, we extend this theory to the setting of multiple tensor products and provide integral representations in concrete function spaces. Our work offers a unified framework for bilinear and multilinear analysis, with a universal constant serving as the fundamental bound.

A Fubini Theorem for Grothendieck Functional Integrals

TL;DR

This work introduces Grothendieck functional integrals as a universal, measure-free framework for bounded linear functionals on projective tensor products, controlled by Grothendieck's constant and equivalent to bilinear forms via a Hilbert-space representation. It establishes a Hilbert-space factorization and an abstract Fubini theorem, then extends the theory to multilinear settings with a multilinear Fubini theorem. The authors provide concrete realizations in and spaces, and demonstrate applications to pseudodifferential-type operators, including kernel representations, norm bounds, and spectral properties governed by . The framework offers a unifying, dimension-free approach to bilinear and multilinear analysis with potential impact on operator theory, PDEs, and numerical methods. Overall, the paper presents a universal, structurally rich method for analyzing integral-type functionals via a fixed constant , enabling efficient exchange of integration order and explicit operator representations across diverse function spaces.

Abstract

This paper systematically studies the subset of continuous linear functionals on the projective tensor product of Banach spaces whose norms are bounded by Grothendieck's constant . We term such functionals Grothendieck functional integrals. The integral is defined as a linear functional on the projective tensor product space that satisfies the boundedness condition , where denotes Grothendieck's constant. We prove that such integrals admit a Hilbert space representation theorem and establish the corresponding abstract Fubini theorem to demonstrate that the order of integration may be interchanged. Furthermore, we extend this theory to the setting of multiple tensor products and provide integral representations in concrete function spaces. Our work offers a unified framework for bilinear and multilinear analysis, with a universal constant serving as the fundamental bound.
Paper Structure (26 sections, 14 theorems, 54 equations)

This paper contains 26 sections, 14 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Projective tensor products
  4. Grothendieck's inequality
  5. Grothendieck Functional Integrals
  6. Definition and basic properties
  7. Hilbert Space Representation
  8. Basic examples
  9. Fubini theorems
  10. Partial integration operators
  11. Abstract Fubini theorem
  12. Operator form of Fubini theorem
  13. Multiple Grothendieck integrals and Fubini theorems
  14. Multilinear definition
  15. Multiple Fubini theorem
  16. ...and 11 more sections

Key Result

Theorem 2.1

There exists a universal constant $K_G$ (with $K_G^\mathbb{R}\approx1.778$ for the real field and $K_G^\mathbb{C}\approx1.338$ for the complex field) such that for arbitrary Banach spaces $E,F$ and any continuous bilinear form $\varphi: E \times F \to \mathbb{K}$, there exist a Hilbert space $H$ and and $\|A\| \|B\| \leq K_G \|\varphi\|$. In particular, if $\|\varphi\| \leq 1$, then there exists s

Theorems & Definitions (41)

  • Theorem 2.1: Grothendieck's inequality
  • Definition 3.1: Grothendieck functional integral
  • Proposition 3.2: Correspondence with Bilinear Forms
  • proof
  • Theorem 3.3: Hilbert Space Representation Theorem
  • proof
  • Remark 3.4
  • Example 3.5
  • Example 3.6
  • Definition 4.1: Partial integration
  • ...and 31 more