The Paley-Wiener Theorem for the Hua System

摘要

E. Damek, A. Hulanicki, and R. Penney (J. Funct. Anal., in press) studied a canonical system of differential equations (the Hua system) denoted HJK which is definable on any Kahlerian manifold M. Functions annihilated by this system are called "Hua-harmonic." In the case where M is a bounded homogeneous domain in C~n with 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 Koranyi 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-Szego 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 in L~(infinity) if and only if the domain is symmetric. We also show that an L~2 function is the boundary function for a Hua-harmonic function if and only if its Fourier transform is supported in a certain (typically nonconvex) cone. This cone is characterized in terms of the Fourier transformation of the Cauchy-Szego Poisson Kernel.