On Poisson transform for spinors

Let $(τ,V_τ)$ be a spinor representation of $\mathrm{Spin}(n)$ and let $(σ,V_σ)$ be a spinor representation of $\mathrm{Spin}(n-1)$ that occurs in the restriction $τ_{\mid \mathrm{Spin}(n-1)}$. We consider the real hyperbolic space $H^n(\mathbb R)$ as the rank one homogeneous space $\mathrm{Spin}_0(1,n)/\mathrm{Spin}(n)$ and the spinor bundle $ΣH^n(\mathbb R)$ over $H^n(\mathbb R)$ as the homogeneous bundle $\mathrm{Spin}_0(1,n)\times_{\mathrm{Spin}(n)} V_τ$. Our aim is to characterize eigenspinors of the algebra of invariant differential operators acting on $ΣH^n(\mathbb R)$ which can be written as the Poisson transform of $L^p$-sections of the bundle $\mathrm{Spin}(n)\times_{\mathrm{Spin}(n-1)} V_σ$ over the boundary $S^{n-1}\simeq \mathrm{Spin}(n)/\mathrm{Spin}(n-1)$ of $H^n(\mathbb R)$, for $1<p<\infty$.

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