Abstract
Consider the generalized absolute value function defined by
.
Further, consider the n-th order divided difference function a[n]: ℝn+1 → ℂ and let 1 < p1, …, pn < ∞ be such that \(\sum\nolimits_{l = 1}^n {p_l^{- 1} = 1} \). Let \({{\cal S}_{{p_l}}}\) denote the Schatten-von Neumann ideals and let \({{\cal S}_{1,\infty}}\) denote the weak trace class ideal. We show that for any (n + 1)-tuple A of bounded self-adjoint operators the multiple operator integral \(T_{{a^{[n]}}}^{\bf{A}}\) maps \({{\cal S}_{{p_1}}} \times \cdots \times {{\cal S}_{{p_n}}}\) to \({{\cal S}_{1,\infty}}\) 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 \(T_{{f^{[n]}}}^{\bf{A}}\) from \({{\cal S}_{{p_1}}} \times \cdots \times {{\cal S}_{{p_n}}}\) to \({{\cal S}_1}\) 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.
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The authors are grateful to the referee for detailed comments which helped to improve the exposition.
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MC is supported by the NWO Vidi grant ‘Non-commutative harmonic analysis and rigidity of operator algebras’, VI.Vidi.192.018.
FS is supported by the ARC Laureate Fellowship.
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Caspers, M., Sukochev, F. & Zanin, D. Weak (1,1) estimates for multiple operator integrals and generalized absolute value functions. Isr. J. Math. 244, 245–271 (2021). https://doi.org/10.1007/s11856-021-2179-0
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DOI: https://doi.org/10.1007/s11856-021-2179-0