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On the best constants of Schur multipliers of higher order divided difference functions

Martijn Caspers, Jesse Reimann

Abstract

Let $f \in C^n(\mathbb{R})$ be such that $\Vert f^{(n)} \Vert_\infty < \infty$. Let $f^{[n]} \in C(\mathbb{R}^{n+1})$ be the $n$th order divided difference. A special case of our main result states that for $1 < p < \infty$ we have \[\Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert \lesssim p^\ast p^n \Vert f^{(n)} \Vert_\infty, \] where $p^\ast = p/(p-1)$ is the Hölder conjugate of $p$ and $T_{f^{[n]}}$ is the multilinear Schur multiplier with symbol $f^{[n]}$. In case of the generalized absolute value map $f(λ) = λ^{n-1} \vert λ\vert, λ\in \mathbb{R}$, we show that \[p^\ast p^{\min(2,n)} \lesssim \Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert.\] This provides an alternative proof to one of the key theorems in the solution of Koplienko's problem on higher order spectral shift [Invent. Math. 193, No. 3, 501-538 (2013)], which is moreover sharp as $p \searrow 1$ (for any $n$) and as $p \to\infty$ (at least if $n = 1,2$).

On the best constants of Schur multipliers of higher order divided difference functions

Abstract

Let be such that . Let be the th order divided difference. A special case of our main result states that for we have \[\Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert \lesssim p^\ast p^n \Vert f^{(n)} \Vert_\infty, \] where is the Hölder conjugate of and is the multilinear Schur multiplier with symbol . In case of the generalized absolute value map , we show that \[p^\ast p^{\min(2,n)} \lesssim \Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert.\] This provides an alternative proof to one of the key theorems in the solution of Koplienko's problem on higher order spectral shift [Invent. Math. 193, No. 3, 501-538 (2013)], which is moreover sharp as (for any ) and as (at least if ).

Paper Structure

This paper contains 21 sections, 21 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Homogeneous sets
  5. Divided differences
  6. Multilinear maps
  7. Schatten classes
  8. Schur multipliers
  9. The Hörmander-Mikhlin-Schur multiplier theorem
  10. A reduction formula for divided differences
  11. Expansions for smooth homogeneous symbols
  12. Decomposition of divided differences
  13. Outline of the proof
  14. Partitions of unity
  15. Auxiliary functions
  16. ...and 6 more sections

Key Result

Theorem A

For every $f \in C^n(\mathbb{R})$ such that $\Vert f^{(n)} \Vert < \infty$ and for every ${1 < p, p_1, \ldots, p_n < \infty}$ with $p^{-1} = \sum_{i=1}^n p_i^{-1}$ we have where $D_n(p,p_1,\dotsc,p_n)<\infty$ is independent of $f$. If $p_1=\dotsc=p_n=np$, then for some $C_n<\infty$ independent of $p$.

Figures (1)

  • Figure 1: From left to right, pictures of the sets: $V_1= \{ (\xi_1, \xi_2, \xi_3) \in \mathbb{R}^3 \mid \vert \xi_2 \vert < 2 \vert \xi_1 \vert, \vert \xi_3 \vert < 2 \vert \xi_1 \vert \},V_2=\{ (\xi_1, \xi_2, \xi_3) \in \mathbb{R}^3 \mid \vert \xi_3 \vert < 2 \vert \xi_2 \vert \},V_3=\{ (\xi_1, \xi_2, \xi_3) \in \mathbb{R}^3 \mid \vert \xi_2 \vert < 2 \vert \xi_3 \vert \}.$We have $U_{\{0,1,2,3\}, 1} =\{ \lambda \in \mathbb{R}^3 \mid (\lambda_1 - \lambda_0, \lambda_2 - \lambda_1, \lambda_3 - \lambda_2) \in V_1 \},U_{\{1,2,3\}, 2} =\{ \lambda \in \mathbb{R}^3 \mid (\lambda_1 - \lambda_0, \lambda_2 - \lambda_1, \lambda_3 - \lambda_2) \in V_2 \},U_{\{1,2,3\}, 3} =\{ \lambda \in \mathbb{R}^3 \mid (\lambda_1 - \lambda_0, \lambda_2 - \lambda_1, \lambda_3 - \lambda_2) \in V_3 \}.$

Theorems & Definitions (56)

  • Theorem A: Upper bound
  • Theorem B: Lower bound
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Lemma 3.1
  • Corollary 3.2
  • ...and 46 more