The Paley–Wiener Theorem for the Hua System

E. Damek, A. Hulanicki, and R. Penney (J. Funct. Anal., in press) studied a canonical system of differential equations (the Hua system) denotedHJKwhich is definable on any Kählerian manifoldM. Functions annihilated by this system are called “Hua-harmonic.” In the case whereMis a bounded homogeneous domain inCnwith its Bergman metric, it was shown that every bounded Hua-harmonic function has a boundary value on the Bergman–Shilov boundary and that the function is reproducible from the Shilov boundary by integration against the reduction of the Poisson kernel for the Laplace–Beltrami operator to the Shilov boundary. This then provided a partial generalization of the results of Johnson and Korányi to the stated context. Significantly, however, no characterization of the resulting space of boundary functions so obtained was given. The current work extends these results in several ways. We show that for a tube domain (i.e., a Siegel domain of type I), the Cauchy–Szegö Poisson kernel also reproduces the Hua-harmonic functions. Since the two kernels agree only in the symmetric case, it follows that the space of boundary functions is dense inL∞if and only if the domain is symmetric. We also show that anL2function is the boundary function for a Hua-harmonic function if and only if its Fourier transform is supported in a certain (typically non-convex) cone. This cone is characterized in terms of the Fourier transformation of the Cauchy–Szegö Poisson kernel.