Commutators on half-spaces

We study the boundedness and compactness of commutators MfIk - IkMf on L-p (R-+(n), dv), where M-f and I-f are defined by M-f[g](x) = f(x)g(x) and I-k[g](x) = integral(H) k(x,y)g(y) dv(y) respectively. If k satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for MfIk - IkMf to be bounded or compact on L-p(R-+(n), dv) for 1 < p < infinity The reproducing kernel of the harmonic Bergman space of H can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.