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Complex Hanner's Inequality for Many Functions

Jonathan Jenkins, Tomasz Tkocz

TL;DR

This work extends Hanner's inequality to arbitrarily many functions by replacing the Rademacher distribution with high-dimensional random vectors uniform on spheres, establishing a dimension-sensitive inequality for $p\ge 2$ that generalizes the classical two-function case. The authors adapt Schechtman’s Hessian/contraction approach by proving the concavity of a 1-homogeneous function $\phi_n$ built from projected sphere vectors, using rotational invariance and unimodality to control the Hessian’s cross-terms. The main result yields a sharp $p$-concavity-type inequality for $d\ge 3$ and $p\ge 2$, with a complementary argument for $d=2$ via complex-analytic tools, and discusses the corresponding implications for complex $L_p$ spaces and unconditional basic sequences. The paper also outlines open ranges for the inequality in low dimensions and links to basic sequence theory, indicating how these probabilistic-type inequalities govern the structure of function spaces under random signings.

Abstract

We establish Hanner's inequality for arbitrarily many functions in the setting where the Rademacher distribution is replaced with higher dimensional random vectors uniform on Euclidean spheres.

Complex Hanner's Inequality for Many Functions

TL;DR

This work extends Hanner's inequality to arbitrarily many functions by replacing the Rademacher distribution with high-dimensional random vectors uniform on spheres, establishing a dimension-sensitive inequality for that generalizes the classical two-function case. The authors adapt Schechtman’s Hessian/contraction approach by proving the concavity of a 1-homogeneous function built from projected sphere vectors, using rotational invariance and unimodality to control the Hessian’s cross-terms. The main result yields a sharp -concavity-type inequality for and , with a complementary argument for via complex-analytic tools, and discusses the corresponding implications for complex spaces and unconditional basic sequences. The paper also outlines open ranges for the inequality in low dimensions and links to basic sequence theory, indicating how these probabilistic-type inequalities govern the structure of function spaces under random signings.

Abstract

We establish Hanner's inequality for arbitrarily many functions in the setting where the Rademacher distribution is replaced with higher dimensional random vectors uniform on Euclidean spheres.
Paper Structure (11 sections, 7 theorems, 37 equations)

This paper contains 11 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction and the main result
  2. Proofs
  3. Overview
  4. Equivalent forms
  5. Hessian
  6. Two-point inequalities
  7. Unimodality and rotational invariance
  8. Proof of Theorem \ref{['thm:main']}
  9. Concluding remarks
  10. Basic sequences
  11. Missing range

Key Result

Theorem 1

Let $d \geq 2$ and let $\xi_1, \xi_2, \dots$ be i.i.d. random vectors uniform on the unit Euclidean sphere $S^{d-1}$ in $\mathbb{R}^d$. Let $p \geq 2$. For every $n \geq 1$ and functions $f_1, \dots, f_n$ in $L_p$, we have Assuming that $d \geq 3$, the reverse inequality holds when $1 \leq p \leq 2$.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Remark 6
  • ...and 3 more