The best constants for operator Lipschitz functions on Schatten classes

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Suppose that f is a Lipschitz function on the real numbers with Lipschitz constant smaller or equal to 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let 1<p<infinity and suppose that x in B(H) is an operator such that the commutator [A, x] is contained in the Schatten class Sₚ. It is proved by the last two authors, that then also [f(A), x] is contained in Sₚ and there exists a constant Cₚ independent of x and f such that || [f(A), x] ||ₚ <= Cₚ || [A,x] ||ₚ. The main result of this paper is to give a sharp estimate for Cₚ in terms of p. Namely, we show that Cₚ p²/(p-1). In particular, this gives the best estimates for operator Lipschitz inequalities. We treat this result in a more general setting. This involves commutators of n self-adjoint operators, for which we prove the analogous result. The case described here in the abstract follows as a special case.