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

Abstract

Consider the generalized absolute value function defined by a(t) = | t| tⁿ⁻¹, t∈ ℝ, n∈ ℕ_{≥ 1}. Further, consider the n-th order divided difference function a^[n]: ℝⁿ⁺¹ → ℂ and let 1 < p₁, …, 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 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.