WEAK TYPE OPERATOR LIPSCHITZ AND COMMUTATOR ESTIMATES FOR COMMUTING TUPLES

Let f : R-d -> R be a Lipschitz function. If B is a bounded self-adjoint operator and if {A(k)}(k=1)(d) are commuting bounded self-adjoint operators such that [A(k), B] is an element of L-1(H), then parallel to[f(A(1), ..., A(d)), B]parallel to(1,infinity) <= c(d)parallel to del(f)parallel to(infinity) max(1 <= k <= d) parallel to[A(k), B]parallel to(1), where c(d) is a constant independent of f, M and A, B and parallel to . parallel to(1,infinity) denotes the weak L-1-norm. If {X-k}(k=1)(d) (respectively, {Y-k}(k=1)(d)) are commuting bounded self-adjoint operators such that X-k - Y-k is an element of L-1 (H), then parallel to f(X-1, ..., X-d) - f(Y-1, ..., Y-d)parallel to(1,infinity) <= c(d)parallel to del(f)parallel to(infinity) max(1 <= k <= d) parallel to X-k - Y-k parallel to(1).