Weak $(1,1)$ estimates for multiple operator integrals and generalized absolute value functions (2004.02145v3)
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 $a{[n]}: \mathbb{R}{n+1} \rightarrow \mathbb{C}$ and let $1 < p_1, \ldots, p_n < \infty$ 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 ${\bf A}$ of bounded self-adjoint operators the multiple operator integral $T_{a{[n]}}{{\bf A}}$ maps $\mathcal{S}{p_1} \times \ldots \times \mathcal{S}{p_n}$ to $\mathcal{S}{1, \infty}$ boundedly with uniform bound in ${\bf A}$. 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 $T{ {\bf A} }{f{[n]}}$ from $\mathcal{S}{p_1} \times \ldots \times \mathcal{S}{p_n}$ 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.