## Abstract

Consider the generalized absolute value function defined by a(t) = | t| t^{n}^{−}^{1}, t∈ ℝ, n∈ ℕ_{≥ 1}. Further, consider the n-th order divided difference function a^{[n]}: ℝ^{n+1} → ℂ and let 1 < p_{1}, …, p_{n} < ∞ be such that ∑l=1npl−1=1. Let Spl denote the Schatten-von Neumann ideals and let S_{1}_{,}_{∞} denote the weak trace class ideal. We show that for any (n + 1)-tuple A of bounded self-adjoint operators the multiple operator integral Ta[n]A maps Sp1×⋯×Spn to S_{1}_{,}_{∞} boundedly with uniform bound in 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 {atf} in this class such boundedness of Tf[n]A from Sp1×⋯×Spn to 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.

Original language | English |
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Pages (from-to) | 245-271 |

Number of pages | 27 |

Journal | Israel Journal of Mathematics |

Volume | 244 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2021 |