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

Abstract

Let $n\in {\mathbb{N}}_{\ge 1}$. Let $1\le p_1,\dots ,p_n<\infty$ and set the Hölder combination $p:=(p_1;\dots ;p_n):={\left({\sum }_{j=1}^n{p}_j^{-1}\right)}^{-1}$. Assume further that $0<p\le 1$ 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 $1\le p_1,\dots ,p_n<\infty$ with $p=(p_1;\dots ;p_n)$.

To appear in International Journal of Mathematics