On the best constants of Schur multipliers of higher order divided difference functions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911530040164352 |
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| author | Caspers, Martijn Reimann, Jesse |
| author_facet | Caspers, Martijn Reimann, Jesse |
| contents | 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 \[p^\ast p^{n} \lesssim \Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert.\] 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the best constants of Schur multipliers of higher order divided difference functions Caspers, Martijn Reimann, Jesse Functional Analysis Classical Analysis and ODEs Operator Algebras 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 \[p^\ast p^{n} \lesssim \Vert T_{f^{[n]}}: S_{np} \times \ldots \times S_{np} \rightarrow S_{p} \Vert.\] 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 $n$. |
| title | On the best constants of Schur multipliers of higher order divided difference functions |
| topic | Functional Analysis Classical Analysis and ODEs Operator Algebras |
| url | https://arxiv.org/abs/2511.06616 |