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

Abstract

Consider the generalized absolute value function defined by $$a(t) = \left| t \right|{t^{n - 1}},\,\,\,\,\,t \in \mathbb{R},n \in {\mathbb{N}_{\ge 1}}.$$.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.