A characterization of the kernel of the Poincaré series operator

Masumoto, Makoto

doi:10.1090/S0002-9947-1987-0876473-5

A characterization of the kernel of the Poincaré series operator
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Trans. Amer. Math. Soc. 300 (1987), 695-704 Request permission

Abstract:

Let $\Gamma$ be a finitely generated Fuchsian group of the first kind acting on the unit disk $\Delta$. The kernel of the Poincaré series operator of the Hardy space ${H^p}, 1 < p < \infty$, onto the Bers space ${A_q}(\Gamma )$ of integrable holomorphic automorphic forms of weight $- 2q, q \in {\mathbf {Z}}, q \geq 2$, on $\Delta$ for $\Gamma$ is characterized in terms of Eichler integrals of order $1 - q$ on $\Delta$ for $\Gamma$.

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