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Sharp affine Sobolev type inequalities via the $\Lp$ Busemann-Petty centroid inequality

Julian Haddad, C. Hugo Jimenez, Marcos Montenegro

TL;DR

The paper develops an elementary, geometry-driven method for sharp affine Sobolev-type inequalities by exploiting the $L_p$ Busemann-Petty centroid inequality. It delivers sharp affine versions of the $L_p$ log-Sobolev, Sobolev, and Gagliardo-Nirenberg inequalities, with explicit equality cases governed by ellipsoidal symmetry and volume-preserving linear maps in $SL_n$, while avoiding dependence on the $L_p$ Minkowski problem or Pólya–Szegő principles. The approach centers on a central convex-analytic inequality (Theorem $\mathrm{main\_ineq}$) and leverages CNV’s sharp Euclidean results alongside Gentil-type log-Sobolev bounds, yielding a unified framework. It also provides an affine $L_\infty$ log-Sobolev variant and a sharpened affine GN inequality, highlighting the geometric content of sharp affine functional inequalities and their extremals.

Abstract

We show that the $\Lp$ Busemann-Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Our approach allows also to characterize directly the corresponding equality cases.

Sharp affine Sobolev type inequalities via the $\Lp$ Busemann-Petty centroid inequality

TL;DR

The paper develops an elementary, geometry-driven method for sharp affine Sobolev-type inequalities by exploiting the Busemann-Petty centroid inequality. It delivers sharp affine versions of the log-Sobolev, Sobolev, and Gagliardo-Nirenberg inequalities, with explicit equality cases governed by ellipsoidal symmetry and volume-preserving linear maps in , while avoiding dependence on the Minkowski problem or Pólya–Szegő principles. The approach centers on a central convex-analytic inequality (Theorem ) and leverages CNV’s sharp Euclidean results alongside Gentil-type log-Sobolev bounds, yielding a unified framework. It also provides an affine log-Sobolev variant and a sharpened affine GN inequality, highlighting the geometric content of sharp affine functional inequalities and their extremals.

Abstract

We show that the Busemann-Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Our approach allows also to characterize directly the corresponding equality cases.

Paper Structure

This paper contains 4 sections, 12 theorems, 87 equations.

Table of Contents

  1. Introduction and previous results
  2. Notations and key tools
  3. The strategy of proofs
  4. Proof of Theorem \ref{['thm:AffineLogSobolev-infty']}

Key Result

Theorem 1.1

Let $n \geq 1$ and $p > 1$. Then for any smooth function $f$ on $\mathbb R^n$ satisfying $\int |f|^pdx=1$, we have where $\mathcal{L}_{n,p}$ is the best log-Sobolev constant given in (log). Moreover, equality holds if and only if for some $\sigma>0$, $x_0 \in\mathbb R^n$ and $A\in SL_n$, for all $x \in \mathbb R^n$, where $SL_n$ denotes the set of $n \times n$-matrices preserving orientation and

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 3.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 9 more