WEAK TYPE COMMUTATOR AND LIPSCHITZ ESTIMATES: RESOLUTION OF THE NAZAROV-PELLER CONJECTURE

Let M be a semi-finite von Neumann algebra and let f : R -> C be a Lipschitz function. If A, B is an element of M are self-adjoint operators such that [A, B] is an element of L-1(M), then parallel to[f (A), B]parallel to(1, infinity) <= C-abs parallel to f'parallel to(infinity)parallel to[A, B]parallel to(1), where C-abs is an absolute constant independent of f , M and A, B and parallel to.parallel to(1,infinity) denotes the weak L-1 -norm. If X, Y is an element of M are self-adjoint operators such that X - Y is an element of L-1(M), then parallel to f(X) - f(Y)parallel to(1,infinity )<= C-abs parallel to f'parallel to(infinity)parallel to X-Y parallel to(1). This result resolves a conjecture raised by F. Nazarov and V. Peller implying a couple of existing results in perturbation theory.