Weak $(1,1)$ estimates for multiple operator integrals and generalized absolute value functions

Abstract

Consider the generalized absolute value function defined by \[ a(t) = \vert t \vert t^{n-1}, \qquad t \in \mathbb{R}, n \in \mathbb{N}_{\geq 1}. \] Further, consider the $n$-th order divided difference function $a^{[n]}:{\mathbb{R}}^{n+1}\to \mathbb{C}$ and let $1<p_1,\dots ,p_n<\infty$ be such that ${\sum }_{l=1}^n{p}_l^{-1}=1$. Let ${\mathcal{S}}_{p_l}$ denote the Schatten-von Neumann ideals and let ${\mathcal{S}}_{1,\infty }$ denote the weak trace class ideal. We show that for any $(n+1)$-tuple $\mathbf{A}$ of bounded self-adjoint operators the multiple operator integral $T_{a^{[n]}}^{\mathbf{A}}$ maps ${\mathcal{S}}_{p_1}\times \dots \times {\mathcal{S}}_{p_n}$ to ${\mathcal{S}}_{1,\infty }$ boundedly with uniform bound in $\mathbf{A}$. The same is true for the class of $C^{n+1}$-functions that outside the interval $[-1,1]$ equal $a$. In [CLPST16] it was proved that for a function $f$ in this class such boundedness of $T_{f^{[n]}}^{\mathbf{A}}$ from ${\mathcal{S}}_{p_1}\times \dots \times {\mathcal{S}}_{p_n}$ to ${\mathcal{S}}_1$ may fail, resolving a problem by V. Peller. This shows that the estimates in the current paper are optimal. The proof is based on a new reduction method for arbitrary multiple operator integrals of divided differences.

to appear in Israel Journal of Mathematics