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

Let $f\in C^n(\mathbb{R})$ be such that $\mathrm{\parallel }{f}^{(n)}{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$. 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 $1<p<\mathrm{\infty }$ we have $$\mathrm{\parallel }{T}_{f^{[n]}}:S_{np}\times \dots \times S_{np}\to S_p\mathrm{\parallel }\lesssim p^{\ast }{p}^n\mathrm{\parallel }{f}^{(n)}{\mathrm{\parallel }}_{\mathrm{\infty }},$$ where $p^{\ast }=p\mathrm{/}(p-1)$ 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 $f(\lambda )={\lambda }^{n-1}\mathrm{\mid }\lambda \mathrm{\mid },\lambda \in \mathbb{R}$, we show that $$p^{\ast }{p}^n\lesssim \mathrm{\parallel }{T}_{f^{[n]}}:S_{np}\times \dots \times S_{np}\to S_p\mathrm{\parallel }\mathrm{.}$$ 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 $p\to \mathrm{\infty }$ for any $n$.