Nazarov–Peller weak type Lipschitz conjecture

Let $H$ be a Hilbert space, let $L_1(H)$ denote the trace-class operators on $H$, and let $L_{1,cinfty}(H)$ denote the weak-$L_1$ space of compact operators $T$ whose singular values satisfy $cmu(k,T)=O(1/(k+1))$. For a Lipschitz function $f:\mathbb{R}\to\mathbb{C}$, write $f(A)$ for its functional calculus on a self-adjoint operator $A$. NazarovccPeller conjecture. Whenever $A,Bcin B(H)$ are self-adjoint and $A-Bcin L_1(H)$, one has $f(A)-f(B)cin L_{1,cinfty}(H)$ and

$$\|f(A)-f(B)\|_{1,cinfty}\leq c_{abs}\|f'\|_\infty\|A-B\|_1,$$

for some absolute constant $c_{abs}$. This weak-type estimate was the major open question underlying the Lipschitz estimates discussed in the paper; the paper presents its resolution.