A characterization of the $L^2$-range of the generalized spectral projections related to the Hodge-de Rham Laplacian

Let $H^n(\mathbb{R})$ be the real hyperbolic space. In this paper, we present a characterization of the $L^2$-range of the generalized spectral projections on the bundle of differential forms over $H^n(\mathbb{R})$. As an underlying result we show a characterization of the $L^2$-range of the Poisson transform on the bundle of differential forms on the boundary $\mathrm{\partial }{H}^n(\mathbb{R})$. This gives a positive answer to a conjecture of Strichartz on differential forms.