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

Martijn Caspers*, Fedor Sukochev, Dmitriy Zanin

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

Consider the generalized absolute value function defined by a(t) = | t| tn1, t∈ ℝ, n∈ ℕ≥ 1. Further, consider the n-th order divided difference function a[n]: ℝn+1 → ℂ and let 1 < p1, …, pn < ∞ be such that ∑l=1npl−1=1. Let Spl denote the Schatten-von Neumann ideals and let S1, 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 S1, boundedly with uniform bound in A. The same is true for the class of Cn+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 S1 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 languageEnglish
Pages (from-to)245-271
Number of pages27
JournalIsrael Journal of Mathematics
Volume244
Issue number1
DOIs
Publication statusPublished - 2021

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