Weak \((1,1)\) estimates for multiple operator integrals and generalized absolute value functions (Q2055273)

Let the generalized absolute value function be defined by \[ \begin{array}{cc} a(t)=\left\vert t\right\vert t^{n-1}, & t\in \mathbb{R},\ n\geq 1 \end{array}, \] and let \(a^{[ n] }:\mathbb{R}^{n+1}\rightarrow \mathbb{C}\) denote the \(n\)-th order divided difference function. In this paper, the authors show that, for any \((n+1)\)-tuple \(\mathbf{A}\) of bounded self-adjoint operators, the multiple operator integral \(T_{a^{[ n] }}^{A}\) maps \(\mathcal{S}_{p_{1}}\times \dots\times \mathcal{S}_{p_{n}}\) to \(\mathcal{S} _{1,\infty }\) boundedly with uniform bound in \(\mathbf{A}\). Here, \(\mathcal{S }_{p_{1}},\dots,\mathcal{S}_{p_{n}}\) are the Schatten-von Neumann ideals with \(\sum\nolimits_{l=1}^{n}\frac{1}{p_{l}}=1\) and \(\mathcal{S}_{1,\infty}\)\ is the weak trace class ideal. The same is true for the class of \(C^{n+1}\) -functions that outside the interval \([ -1,1] \) equal \(a\).