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

TL;DR

This paper establishes bounds on the constants of Schur multipliers associated with higher order divided difference functions, providing sharp estimates that are crucial for spectral shift problems in operator theory.

Contribution

It offers new sharp bounds on the norms of multilinear Schur multipliers for higher order divided differences, advancing understanding in spectral shift function analysis.

Findings

Bounded the Schur multiplier norms by p* p^n times the infinity norm of the nth derivative.

Provided an alternative proof for a key theorem in Koplienko's spectral shift problem.

Showed the bounds are sharp as p approaches 1 and infinity for any order n.

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 H\"older conjugate of $p$ and $T_{f^{[n]}}$ is the multilinear Schur multiplier with symbol $f^{[n]}$. In case of the generalized absolute value map $f(\lambda) = \lambda^{n-1} \vert \lambda \vert, \lambda \in \mathbb{R}$, we show that \[p^\ast p^{n} \lesssim \Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert.\] 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 $p \searrow 1$ and as $p \to\infty$ for any $n$.