  We study the Poisson transform for differential forms on the real hyperbolic
space $\mathbb H^n$. For $1<r<\infty$, we prove that the Poisson transform is
an isomorphism from the space of $L^r$ $q$-differential forms on the boundary
$\partial \mathbb H^n$ onto a Hardy-type subspace of $p$-eigenforms of the de
Rham-Hodge Laplacian, for $0\leq p<\frac{n-1}{2}$ and $q=p-1, p$.
