Higher order perturbation estimates in quasi-Banach Schatten spaces through wavelets

Abstract: Let $n \in \mathbb{N}{\geq 1}$. Let $1 \leq p_1, \ldots, p_n < \infty$ and set the H\"older 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 H\"older 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 $p = (p_1; \ldots; p_n)$.