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Interpolation of derivatives and ultradifferentiable regularity

Armin Rainer, Gerhard Schindl

Abstract

Interpolation inequalities for $C^m$ functions allow to bound derivatives of intermediate order $0 < j<m$ by bounds for the derivatives of order $0$ and $m$. We review various interpolation inequalities for $L^p$-norms ($1 \le p \le \infty$) in arbitrary finite dimensions. They allow us to study ultradifferentiable regularity by lacunary estimates in a comprehensive way, striving for minimal assumptions on the weights.

Interpolation of derivatives and ultradifferentiable regularity

Abstract

Interpolation inequalities for functions allow to bound derivatives of intermediate order by bounds for the derivatives of order and . We review various interpolation inequalities for -norms () in arbitrary finite dimensions. They allow us to study ultradifferentiable regularity by lacunary estimates in a comprehensive way, striving for minimal assumptions on the weights.
Paper Structure (9 sections, 18 theorems, 84 equations)

This paper contains 9 sections, 18 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Interpolation inequalities
  3. The global setting
  4. The local setting
  5. The Cartan--Gorny inequality
  6. Ultradifferentiable regularity by lacunary estimates
  7. Denjoy--Carleman classes
  8. Other ultradifferentiable classes
  9. Comparison with some recent results

Key Result

Lemma 2.1

[l]lem:minterpol Let $1 \le p \le \infty$ and $m \in \mathbb{N}_{\ge 2}$. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a $C^m$ function. Then, for all $v \in \mathbb S^{n-1}$, if the right-hand side is finite, where $d_v^jf(x) := \partial_t^j f(x+tv)|_{t=0}$.

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • ...and 30 more