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The harmonic analysis of automorphic functions has been extensively 
studied; references to the pertinent parts of this theory are contained
in our monograph.We recall that the Poincar\'e plane $\Pi$, that is the
upper half-plane $$  w= x+iy,\qquad y>0,\eqno(1.1)$$
serves as a model for a non-Euclidean geometry in which the motions are
given by the group $G$ of fractional linear transformations:
$$ w→ {aw+b\over cw+d}\leqno(1.2)$$
where $a,b,c,d$ are real and $$ad-bc=1;\leqno(1.3)$$
$G$ is isomorphic with $SL(2,R)/{\scriptstyle\pm}I$. The Riemannian metric
$${dx↑2+dy↑2\over y↑2}\leqno(1.4)$$
is invariant under this group of motions. The invariant $L↓2$ form is
$$\int\!\int u ↑2\,{dx\,dy\over y↑2}.\leqno(1.5)$$
   
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For $p≥1$, we let $L↑p(\Omega)$ denote the classical Banach space consisting 
of measurable functions on $\Omega$ that are $p$-integrable. The norm in
 $L↑p(\Omega)$
defined by $$\left\leftvv u\right\rightvv↓{L↑p(\Omega)}=\left(\int↓\Omega|u|↑p\,dx
\right)↑{1/p}.\eqno(7.3)$$
For $p=∞$, $L↑∞(\Omega)$ denotes the Banach space of bounded functions on $\Omega$ 
with the norm
$$\left\leftvv u\right\rightvv↓{L↑∞(\Omega)}=\sup↓\Omega |u|.\eqno(7.4)$$
In the following we shall use 
$\left\leftvv u\right\rightvv↓p$ for 
$\left\leftvv u\right\rightvv↓{L↑p(\Omega)}$ when there is no ambiguity.

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