2020/09/17 by McDonald, Edward, Sukochev, Fedor · 2 citations
#47A30 #47B10 #47L20 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2009.08069
We study the class of functions f on ℝ satisfying a Lipschitz estimate in the Schatten ideal Lp for 0 < p ≤ 1. The corresponding problem with p≥ 1 has been extensively studied, but the quasi-Banach range 0 < p < 1 is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class B(1)/(p)(p)/(1-p),p(ℝ) obey the estimate ‖f(A)-f(B)‖p ≤ Cp(‖f'‖_L∞(ℝ)+‖f‖_B(1)/(p)(p)/(1-p),p(ℝ))‖A-B‖p for all bounded self-adjoint operators A and B with A-B∈ Lp. In the case p=1, our methods recover and provide a new perspective on a result of Peller that f ∈ B1∞,1 is sufficient for a function to be Lipschitz in L1. We also provide related Hölder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on ℝ are not Lipschitz in Lp for any 0 < p < 1. This gives counterexamples to a 1991 conjecture of Peller that f ∈ B1/p∞,p(ℝ) is sufficient for f to be Lipschitz in Lp.