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

Consider the generalized absolute value function defined by $$a(t) = \left| t \right|{t^{n - 1}},\,\,\,\,\,t \in \mathbb{R},n {\mathbb{N}_{\ge 1}}.$$ . Further, consider n-th order divided difference a[n]: ℝn+1 → ℂ and let 1 < p1, …, pn ∞ be such that $$\sum\nolimits_{l 1}^n {p_l^{- 1} $$ Let $${{\cal S}_{{p_l}}}$$ denote Schatten-von Neumann ideals S}_{1,\infty}}$$ weak trace class ideal. We show for any (n + 1)-tuple A of bounded self-adjoint operators multiple operator integral $$T_{{a^{[n]}}}^{\bf{A}}$$ maps S}_{{p_1}}} \times \cdots {{\cal S}_{{p_n}}}$$ to boundedly with uniform bound in A. The same is true Cn+1-functions outside interval [−1, 1] equal a. In [CLPST16] it was proved a {atf} this boundedness $$T_{{f^{[n]}}}^{\bf{A}}$$ from S}_1}$$ may fail, resolving problem V. Peller. This shows estimates current paper are optimal. proof based on new reduction method arbitrary integrals differences.