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

Let $n\in {\mathbb{N}}_{\ge 1}$. Let $1\le p_1,\dots ,p_n<\mathrm{\infty }$ 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, $$1\le (p_2;\dots ;p_n),(p_1;\dots ;p_{n-1})<\mathrm{\infty }\mathrm{.}$$ 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 $\mathrm{\parallel }{f}^{(n)}{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$ we have $$\mathrm{\parallel }{T}_{f^{[n]}}:S_{p_1}\times \dots \times S_{p_n}\to S_p\mathrm{\parallel }\le C(\mathrm{\parallel }{f}^{(n)}{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }f{\mathrm{\parallel }}_{{\dot{B}}_{\frac{p}{1-p},p}^{n-1+\frac{1}{p}}})\mathrm{.}$$ 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 $1\le p_1,\dots ,p_n<\mathrm{\infty }$ with $p=(p_1;\dots ;p_n)$.