Boundedness of Bergman projections on tube domains over light cones

Let Γ be the future light cone in ℝn, and Ω = ℝn + iΓ be the associated tube domain. We prove that the weighted Bergman projection Pv pvf(z) = ∫Ω f(w)Q(z - w̄)-vQ(script T signmw)v-ndw is bounded on Lp(Ω, Qv-n(script T signmw)dw) for 1 + n-2/2(v-1) < p < 1 + 2(v-1)/n-2, where Q denotes the Lorentz quadratic form. This theorem extends previous results by Bekollé and Bonami [BB]. Our proof relies on the analysis of the projection Pv, on mixed norm spaces, which allows us to exploit the oscillation of the Bergman kernel using the Laplace-Fourier transform.