Sharp commutator estimates via harmonic extensions

Abstract

We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations. The commutators we investigate are Jacobians, more generally Coifman–Rochberg–Weiss commutators, Chanillo's commutator with the Riesz potential, Coifman–McIntosh–Meyer, Kato–Ponce–Vega type commutators, the fractional Leibniz rule, and the Da Lio–Rivière three-term commutators. We also give a new limiting L1-estimate for a double commutator of Coifman–Rochberg–Weiss-type, and several intermediate estimates. The beauty of our method is that all those commutator estimates, which are originally proven by various specific methods or by general para-product arguments, can be obtained purely from integration by parts and trace theorems. Another interesting feature is that in all these cases the “cancellation effect” responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way.