# 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
How to Cite This Entry:
Bers space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bers_space&oldid=46029
This article was adapted from an original article by M. Masumoto (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article