On the A_q (Γ) ⊂ B_q (Γ) Conjecture

Abstract

Let Γ be a Fuchsian group acting on the unit disk U : |z| < 1 and let q ≥ 1; temporarily we assume q integral. The holomorphic function f defined in U is called an automorphic form of weight q (or degree -2q) if

$$ f(Az)A'(z)^q = f(z),z\varepsilon U,A\varepsilon \Gamma . $$

(1)

Let R be a normal polygon (fundamental region) for Γ in U, The automorphic form f is called integrable, and we write f ε A_q (Γ), if

$$ \parallel f\parallel _1 = \smallint _R \smallint (1 - |z|^2 )^{q - 2} |f(z)|dxdy < \infty ; $$

(2)

it is called bounded, and we write f ε B_q (Γ), if

$$ \parallel f\parallel _\infty = \mathop {\sup }\limits_{z\varepsilon U} (1 - |z|^2 )^q |f(z)| < \infty . $$

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These are the well-known Bers’ spaces; see for example [1]. It has been conjectured that

$$ A_q (\Gamma ) \subset B_q (\Gamma ) $$

(4)

for all Fuchsian groups Γ. Though this conjecture is still unsettled, it has been verified when Γ is finitely generated. The situation when Γ is of the first kind being essentially trivial, we state the result as follows :

THEOREM. If Γ is a finitely generated group of the second kind, then

$$ A_q (\Gamma ) \subset B_q (\Gamma ). $$

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