Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918178080161792 |
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| author | Caspers, Martijn Huisman, Emiel |
| author_facet | Caspers, Martijn Huisman, Emiel |
| contents | Let $n \in \mathbb{N}_{\geq 1}$. Let $1 \leq p_1, \ldots, p_n < \infty$ and set the Hölder combination $p := (p_1; \ldots ; p_n) := \left( \sum_{j=1}^n p_j^{-1} \right)^{-1}$. Assume further that $0 < p \leq 1$ and that for the Hölder combinations of $p_2$ to $p_n$ and $p_1$ to $p_{n-1}$ we have,
\[
1 \leq (p_2; \ldots ; p_n), (p_1; \ldots ; p_{n-1}) < \infty.
\]
Then there exists a constant $C> 0$ such that for every $f \in C^n(\mathbb{R}) \cap \dot{B}_{\frac{p}{1-p}, p}^{n-1 + \frac{1}{p}}$ with $\Vert f^{(n)} \Vert_\infty < \infty$ we have \[
\Vert T_{f^{[n]}}: S_{p_1} \times \ldots \times S_{p_n} \rightarrow S_p \Vert \leq C ( \Vert f^{(n)} \Vert_\infty + \Vert f \Vert_{\dot{B}_{\frac{p}{1-p}, p}^{n-1 + \frac{1}{p}}}). \]
Here $S_q$ is the Schatten von Neumann class, $\dot{B}_{p,q}^s$ the homogeneous Besov space, and $T_{f^{[n]}}$ is the multilinear Schur multiplier of the $n$-th order divided difference function. In particular, our result holds for $p=1$ and any $1 \leq p_1, \ldots, p_n < \infty$ with $p = (p_1; \ldots; p_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06558 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets Caspers, Martijn Huisman, Emiel Functional Analysis Operator Algebras Let $n \in \mathbb{N}_{\geq 1}$. Let $1 \leq p_1, \ldots, p_n < \infty$ and set the Hölder combination $p := (p_1; \ldots ; p_n) := \left( \sum_{j=1}^n p_j^{-1} \right)^{-1}$. Assume further that $0 < p \leq 1$ and that for the Hölder combinations of $p_2$ to $p_n$ and $p_1$ to $p_{n-1}$ we have, \[ 1 \leq (p_2; \ldots ; p_n), (p_1; \ldots ; p_{n-1}) < \infty. \] Then there exists a constant $C> 0$ such that for every $f \in C^n(\mathbb{R}) \cap \dot{B}_{\frac{p}{1-p}, p}^{n-1 + \frac{1}{p}}$ with $\Vert f^{(n)} \Vert_\infty < \infty$ we have \[ \Vert T_{f^{[n]}}: S_{p_1} \times \ldots \times S_{p_n} \rightarrow S_p \Vert \leq C ( \Vert f^{(n)} \Vert_\infty + \Vert f \Vert_{\dot{B}_{\frac{p}{1-p}, p}^{n-1 + \frac{1}{p}}}). \] Here $S_q$ is the Schatten von Neumann class, $\dot{B}_{p,q}^s$ the homogeneous Besov space, and $T_{f^{[n]}}$ is the multilinear Schur multiplier of the $n$-th order divided difference function. In particular, our result holds for $p=1$ and any $1 \leq p_1, \ldots, p_n < \infty$ with $p = (p_1; \ldots; p_n)$. |
| title | Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2509.06558 |