TY - JOUR
T1 - Weak (1,1) estimates for multiple operator integrals and generalized absolute value functions
AU - Caspers, Martijn
AU - Sukochev, Fedor
AU - Zanin, Dmitriy
N1 - Accepted author manuscript
PY - 2021
Y1 - 2021
N2 - 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, …, 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.
AB - 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, …, 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.
UR - http://www.scopus.com/inward/record.url?scp=85113171877&partnerID=8YFLogxK
U2 - 10.1007/s11856-021-2179-0
DO - 10.1007/s11856-021-2179-0
M3 - Article
AN - SCOPUS:85113171877
SN - 0021-2172
VL - 244
SP - 245
EP - 271
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -