A fast numerical method for evaluation of Calderَn commutators

We describe a methodology for fast evaluation of multilinear operators that are generated by a rapidly computable nonlinear operator. We illustrate this idea by developing a simple numerical algorithm for the fast evaluation of Calderón commutators of all orders,Cnf(x)=p.v.∫−∞∞ (A(x)−A(y))n(x−y)n+1 f(y) dy,n=1,2,… . The method is based on a representation of the commutators as derivatives of a one parameter family of real-valued versions of Cauchy integrals. We include numerical experiments for the first two commutators. Additionally, we consider the Dirichlet problem for the Laplacian in the unbounded region above the graph of a function. We demonstrate that Calderón commutators appear as building blocks of the functional coefficients of a perturbative solution for this problem.