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

TL;DR

This paper provides new bounds and proofs for the boundedness of bilinear Schur multipliers related to second order divided difference functions, sharpening previous asymptotic bounds and establishing optimality in certain cases.

Contribution

The authors introduce a new proof technique using bilinear transference and the Hörmander-Mikhlin-Schur multiplier theorem, significantly improving bounds for second order divided difference functions.

Findings

Established upper bounds for Schur multipliers involving second order divided differences.

Proved lower bounds that match the upper bounds asymptotically, demonstrating optimality.

Sharpened the understanding of the constants involved in the boundedness of these multipliers.

Abstract

We give a new proof of the boundedness of bilinear Schur multipliers of second order divided difference functions, as obtained earlier by Potapov, Skripka and Sukochev in their proof of Koplienko's conjecture on the existence of higher order spectral shift functions. Our proof is based on recent methods involving bilinear transference and the H\"ormander-Mikhlin-Schur multiplier theorem. Our approach provides a significant sharpening of the known asymptotic bounds of bilinear Schur multipliers of second order divided difference functions. Furthermore, we give a new lower bound of these bilinear Schur multipliers, giving again a fundamental improvement on the best known bounds obtained by Coine, Le Merdy, Potapov, Sukochev and Tomskova. More precisely, we prove that for $f \in C^2(\mathbb{R})$ and $1 < p, p_1, p_2 < \infty$ with $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$ we have \[ \Vert M_{f^{[2]}}: S_{p_1} \times S_{p_2} \rightarrow S_p \Vert \lesssim \Vert f'' \Vert_\infty D(p, p_1, p_2), \] where the constant $D(p, p_1, p_2)$ is specified in Theorem 7.1 and $D(p, 2p, 2p) \approx p^4 p^\ast$ with $p^\ast$ the H\"older conjugate of $p$. We further show that for $f(\lambda) = \lambda \vert \lambda \vert$, $\lambda \in \mathbb{R}$, for every $1 < p < \infty$ we have \[ p^2 p^\ast \lesssim \Vert M_{f^{[2]}}: S_{2p} \times S_{2p} \rightarrow S_p \Vert. \] Here $f^{[2]}$ is the second order divided difference function of $f$ with $M_{f^{[2]}}$ the associated Schur multiplier. In particular it follows that our estimate $D(p, 2p, 2p)$ is optimal for $p \searrow 1$.