Bers space

A complex Banach space of holomorphic automorphic forms introduced by L. Bers (1961). Let $ D $ be an open set of the Riemann sphere $ {\widehat{\mathbf C} } = \mathbf C \cup \{ \infty \} $ whose boundary consists of more than two points. Then $ D $ carries a unique complete conformal metric $ \lambda ( z ) | {dz } | $ on $ D $ with curvature $ - 4 $, known as the hyperbolic metric on $ D $. Let $ G $ be a properly discontinuous group of conformal mappings of $ D $ onto itself (cf. also Kleinian group; Conformal mapping). Typical examples of $ G $ are Kleinian groups (cf. also Kleinian group), that is, a group of Möbius transformations (cf. also Fractional-linear mapping) of $ {\widehat{\mathbf C} } $ acting properly discontinuously on an open set of $ {\widehat{\mathbf C} } $. By the conformal invariance, the hyperbolic area measure $ \lambda ( z ) ^ {2} dx dy $( $ z = x + iy $) on $ D $ is projected to an area measure $ d \mu $ on the orbit space $ D/G $. In other words, let $ d \mu ( w ) = \lambda ( z ) ^ {2} dx dy $, $ w = \pi ( z ) $, where $ \pi : D \rightarrow {D/G } $ is the natural projection.

Fix an integer $ q \geq 2 $. A holomorphic function $ \varphi $ on $ D $ is called an automorphic form of weight $ - 2q $ for $ G $ if $ ( \varphi \circ g ) \cdot ( g ^ \prime ) ^ {q} = \varphi $ for all $ g \in G $. Then $ \lambda ^ {- q } | \varphi | $ is invariant under the action of $ G $ and hence may be considered as a function on $ D/G $. The Bers space $ A _ {q} ^ {p} ( D,G ) $, where $ 1 \leq p \leq \infty $, is the complex Banach space of holomorphic automorphic forms $ \varphi $ of weight $ - 2q $ on $ D $ for $ G $ such that the function $ \lambda ^ {- q } | \varphi | $ on $ D/G $ belongs to the space $ L _ {p} $ with respect to the measure $ \mu $. The norm in $ A _ {q} ^ {p} ( D,G ) $ is thus given by

$$ \left \| \varphi \right \| = ( {\int\limits \int\limits } _ {D/G } {\lambda ^ {- pq } \left | \varphi \right | ^ {p} } {d \mu } ) ^ {1/p } $$

if $ 1 \leq p < \infty $, and

$$ \left \| \varphi \right \| = \sup _ {D/G } \lambda ^ {- q } \left | \varphi \right | $$

if $ p = \infty $. Automorphic forms in $ A _ {q} ^ {p} ( D,G ) $ are said to be $ p $- integrable if $ 1 \leq p < \infty $, and bounded if $ p = \infty $. When $ G $ is trivial, $ A _ {q} ^ {p} ( D,G ) $ is abbreviated to $ A _ {q} ^ {p} ( D ) $. Note that $ A _ {q} ^ \infty ( D,G ) $ is isometrically embedded as a subspace of $ A _ {q} ^ \infty ( D ) $.

Some properties of Bers spaces.

1) Let $ {1 / p } + {1 / {p ^ \prime } } = 1 $. The Petersson scalar product of $ \varphi \in A _ {q} ^ {p} ( D,G ) $ and $ \psi \in A _ {q} ^ {p ^ \prime } ( D,G ) $ is defined by

$$ \left ( \varphi , \psi \right ) = {\int\limits \int\limits } _ {D/G } {\lambda ^ {- 2q } \varphi {\overline \psi \; } } {d \mu } . $$

If $ 1 \leq p < \infty $, then the Petersson scalar product establishes an anti-linear isomorphism of $ A _ {q} ^ {p ^ \prime } ( D,G ) $ onto the dual space of $ A _ {q} ^ {p} ( D,G ) $, whose operator norm is between $ ( q - 1 ) ( 2q - 1 ) ^ {- 1 } $ and $ 1 $.

2) The Poincaré (theta-) series of a holomorphic function $ f $ on $ D $ is defined by

$$ \Theta f = \sum _ {g \in G } ( f \circ g ) \cdot ( g ^ \prime ) ^ {q} $$

whenever the right-hand side converges absolutely and uniformly on compact subsets of $ D $( cf. Absolutely convergent series; Uniform convergence). Then $ \Theta f $ is an automorphic form of weight $ - 2q $ on $ D $ for $ G $. Moreover, $ \Theta $ gives a continuous linear mapping of $ A _ {q} ^ {1} ( D ) $ onto $ A _ {q} ^ {1} ( D,G ) $ of norm at most $ 1 $. For every $ \varphi \in A _ {q} ^ {p} ( D,G ) $ there exists an $ f \in A _ {q} ^ {p} ( D ) $ with $ \| f \| \leq ( 2q - 1 ) ( q - 1 ) ^ {- 1 } \| \varphi \| $ such that $ \varphi = \Theta f $.

3) Let $ B $ be the set of branch points of the natural projection $ \pi $. Assume that: i) $ D/G $ is obtained from a (connected) closed Riemann surface of genus $ g $ by deleting precisely $ m $ points; and ii) $ \pi ( B ) $ consists of exactly $ n $ points $ p _ {1} \dots p _ {n} $( possibly, $ m = 0 $ or $ n =0 $). For each $ k = 1 \dots n $, let $ \nu _ {k} $ be the common multiplicity of $ \pi $ at points of $ \pi ^ {- 1 } ( p _ {k} ) $. Then $ A _ {q} ^ {p} ( D,G ) = A _ {q} ^ \infty ( D,G ) $ for $ 1 \leq p \leq \infty $ and

$$ { \mathop{\rm dim} } A _ {q} ^ \infty ( D,G ) = $$

$$ = ( 2q - 1 ) ( g - 1 ) + ( q - 1 ) m + \sum _ {k =1 } ^ { n } \left [ q \left ( 1 - { \frac{1}{\nu _ {k} } } \right ) \right ] , $$

where $ [ x ] $ denotes the largest integer that does not exceed $ x $.

4) Consider the particular case where $ D $ is the unit disc. Then $ G $ is a Fuchsian group and $ \lambda ( z ) = ( 1 - | z | ^ {2} ) ^ {- 1 } $. It had been conjectured that $ A _ {q} ^ {1} ( D,G ) \subset A _ {q} ^ \infty ( D,G ) $ for any $ G $, until Ch. Pommerenke [a6] constructed a counterexample. In [a5] D. Niebur and M. Sheingorn characterized the Fuchsian groups $ G $ for which the inclusion relation holds. In particular, if $ G $ is finitely generated, then $ A _ {q} ^ {1} ( D,G ) \subset A _ {q} ^ \infty ( D,G ) $.

5) Let $ G $ be a Fuchsian group acting on the unit disc $ D $. It also preserves $ D ^ {*} = {\widehat{\mathbf C} } \setminus {\overline{D}\; } $, the outside of the unit circle. If $ f $ is conformal on $ D ^ {*} $ and can be extended to a quasi-conformal mapping of $ {\widehat{\mathbf C} } $ onto itself such that $ f \circ g \circ f ^ {- 1 } $ is a Möbius transformation for each $ g \in G $, then its Schwarzian derivative

$$ Sf = { \frac{f ^ {\prime \prime \prime } }{f ^ \prime } } - { \frac{3}{2} } \left ( { \frac{f ^ {\prime \prime } }{f ^ \prime } } \right ) ^ {2} $$

belongs to $ A _ {2} ^ \infty ( D ^ {*} ,G ) $ with $ \| {Sf } \| \leq 6 $. Moreover, the set of all such Schwarzian derivatives constitutes a bounded domain in $ A _ {2} ^ \infty ( D ^ {*} ,G ) $ including the open ball of radius $ 2 $ centred at the origin. This domain can be regarded as a realization of the Teichmüller space $ T ( G ) $ of $ G $, and the injection of $ T ( G ) $ into $ A _ {2} ^ \infty ( D ^ {*} ,G ) $ induced by the Schwarzian derivative is referred to as the Bers embedding.

References

[a1]	I. Kra, "Automorphic forms and Kleinian groups" , Benjamin (1972)
[a2]	J. Lehner, "Discontinuous groups and automorphic functions" , Amer. Math. Soc. (1964)
[a3]	J. Lehner, "Automorphic forms" W.J. Harvey (ed.) , Discrete Groups and Automorphic Functions , Acad. Press (1977) pp. 73–120
[a4]	S. Nag, "The complex analytic theory of Teichmüller spaces" , Wiley (1988)
[a5]	D. Niebur, M. Sheingorn, "Characterization of Fuchsian groups whose integrable forms are bounded" Ann. of Math. , 106 (1977) pp. 239–258
[a6]	Ch. Pommerenke, "On inclusion relations for spaces of automorphic forms" W.E. Kirwan (ed.) L. Zalcman (ed.) , Advances in Complex Function Theory , Lecture Notes in Mathematics , 505 , Springer (1976) pp. 92–100