论文标题
多个运算符积分和广义绝对价值功能的弱$（1,1）$估计值
Weak $(1,1)$ estimates for multiple operator integrals and generalized absolute value functions
论文作者
论文摘要
考虑由\ [a（t）= \ vert t \ vert t^{n-1}定义的广义绝对值函数，\ qquad t \ in \ mathbb {r}，n \ in \ in \ 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 $\sum_{l=1}^n p_l^{-1} = 1$.令$ \ Mathcal {s} _ {p_l} $表示Schatten-von neumann理想，让$ \ Mathcal {s} _ {1，\ infty} $表示弱跟踪类理想。我们表明，对于任何$（n+1）$ - tuple $ {\ bf a} $的有界自动接合运算符的多个操作员积分$ t_ {a^{a^{[n]}}}^{\ bf a} $ maps $ maps $ \ \ \ mathcal {s} _ {s} _ {p_1} _ {p_1} _ {p_1} \ times \ ldoest \ Mathcal {s} _ {p_n} $ to $ \ MATHCAL {S} _ {1，\ infty} $在$ {\ bf a} $中均匀地绑定。 $ c^{n+1} $的类也是如此，在间隔$ [ - 1，1] $等于$ a $之外的功能。在[clpst16]中，证明了此类$ f $的$ t^{{\ bf a}} _ {f^{[n]}} $的有限性，来自$ \ m artercal {s}} $ \ MATHCAL {S} _ {1} $可能会失败，从V. Peller解决问题。这表明当前论文中的估计值是最佳的。该证明是基于一种新的还原方法，用于任意分裂差异的多个操作员积分。
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.
