Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets
Martijn Caspers, Emiel Huisman
TL;DR
The paper establishes sharp bounds for higher-order divided differences as multilinear Schur multipliers acting on Schatten spaces in the quasi-Banach range $0<p\leq 1$. Using a wavelet-based decomposition and Besov regularity, it proves that for $f\in C^n(\mathbb{R})\cap \dot B_{\frac{p}{1-p},p}^{n-1+\frac{1}{p}}$ with $\|f^{(n)}\|_\infty<\infty$, the multilinear symbol $f^{[n]}$ induces a bounded operator $T_{f^{[n]}}:S_{p_1}\times\cdots\times S_{p_n}\to S_p$, with a norm controlled by $\|f^{(n)}\|_\infty$ and the Besov norm. The approach combines diagonal and block-diagonal wavelet components via MOI techniques and the Potapov–Skripka–Sukochev theorem, and then interpolates these estimates through induction on the order $n$. A key novelty is the first genuinely nonlinear multilinear bound in the quasi-Banach Schatten setting, extending prior linear and bilinear results to arbitrary order. The results have implications for higher-order Fréchet differentiability of Schatten norms and noncommutative perturbation theory, while also outlining open questions about relaxing exponent constraints and extending to broader Besov scales.
Abstract
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 $p = (p_1; \ldots; p_n)$.