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Gagliardo-Nirenberg inequality with Hölder norms

Mengxia Dong

TL;DR

This paper extends the classical Gagliardo-Nirenberg inequality to a unified framework that combines Sobolev and Hölder spaces via the X^p scale. It introduces an interpolation lemma bridging Lebesgue and Hölder norms, and uses it to prove a generalized GN inequality in which Lebesgue norms are replaced by Hölder norms under suitable parameter conditions, namely $\|D^l u\|_{X^q} \le C \|D^k u\|_{X^p}^{\theta} \|u\|_{X^r}^{1-\theta}$ with $\frac{1}{q}-\frac{l}{n} = \theta\left(\frac{1}{p}-\frac{k}{n}\right)+(1-\theta)\left(\frac{1}{r}\right)$ and $l\le k$. The approach recovers and extends known results (Sobolev, Morrey, Kufner-Wannebo, Molchanova-Soudský) as special cases and broadens the admissible parameter range. The outcome provides a flexible analytic tool for PDE regularity theory by enabling interpolation between derivatives and functions in a space that intermixes integrability and Hölder regularity.

Abstract

The classical Gagliardo-Nirenberg inequality, known as an interpolation inequality, involves Lebesgue norms of functions and their derivatives. We established an interpolation lemma to connect Lebesgue and Hölder spaces, thus extending the Gagliardo-Nirenberg inequality. This extension involved substituting arbitrary Sobolev norms with appropriate Hölder norms, allowing for a wider range of applicable parameters in the inequality.

Gagliardo-Nirenberg inequality with Hölder norms

TL;DR

This paper extends the classical Gagliardo-Nirenberg inequality to a unified framework that combines Sobolev and Hölder spaces via the X^p scale. It introduces an interpolation lemma bridging Lebesgue and Hölder norms, and uses it to prove a generalized GN inequality in which Lebesgue norms are replaced by Hölder norms under suitable parameter conditions, namely with and . The approach recovers and extends known results (Sobolev, Morrey, Kufner-Wannebo, Molchanova-Soudský) as special cases and broadens the admissible parameter range. The outcome provides a flexible analytic tool for PDE regularity theory by enabling interpolation between derivatives and functions in a space that intermixes integrability and Hölder regularity.

Abstract

The classical Gagliardo-Nirenberg inequality, known as an interpolation inequality, involves Lebesgue norms of functions and their derivatives. We established an interpolation lemma to connect Lebesgue and Hölder spaces, thus extending the Gagliardo-Nirenberg inequality. This extension involved substituting arbitrary Sobolev norms with appropriate Hölder norms, allowing for a wider range of applicable parameters in the inequality.
Paper Structure (7 sections, 7 theorems, 109 equations)

This paper contains 7 sections, 7 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Sobolev Inequalities
  3. Interpolation Lemma
  4. Proof of Theorem \ref{['TheoremGagliardoNirenbergInequalityGeneral']}
  5. Initial Inequality
  6. Induction
  7. Interpolation

Key Result

Theorem 1.1

Let $p,q,r\in(-\infty,0)\cup[1,+\infty)$, $k,l\in\mathbb{N}^+$, $k>l$ and $\frac{n}{p}\notin\{1,\cdots,k-l\}$, then for all $u\in C_0^\infty(\mathbb{R}^n)$ with $u\in X^{k,p}(\mathbb{R}^n)\cap X^r(\mathbb{R}^n)$, we have $u\in X^{l,q}(\mathbb{R}^n)$ and there exists a constant $C$ independent of $u$ where for all $\theta$ in the interval

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 7 more