Let $X = SL\_2(\mathbb{Z}) \backslash \mathbb{H}$ be the modular surface. Consider a basis of $L^2$-normalized Hecke-Maass cusps forms $\phi\_j$ on $X$ with $-\Delta$-eigenvalue $\lambda\_j$. Hejhal-Rackner (1991, on the Topography of Maass forms) seems to claim that for any fixed set $A \subset X$ with finite measure,
$$\int\_A \phi\_j (z) d\mu z \to 0$$
as $\lambda\_j \to \infty$.


I am probably overlooking something, but is there a simple proof of this? One of the even/odd Maass forms case should be trivial by sign considerations, but unless I am missing something the other case does not seem as apparent.



Mean value of Maass forms
This is an analog of the Riemann-Lebesgue lemma. Let $f$ be a smooth function on $X$. Then by self-adjointness of the Laplacian 
$$ 
\lambda\_j \langle \phi\_j, f \rangle = \langle \Delta \phi\_j ,f \rangle = \langle \phi\_j, \Delta f\rangle ,
$$ 
and by Cauchy-Schwarz this is bounded in size by $\Vert \phi\_j \Vert \Vert \Delta f\Vert$. Therefore $\langle \phi\_j, f\rangle$ goes to zero as $\lambda\_j$ gets large. Now approximate the characteristic function $\chi\_A$ of the set $A$ by suitable smooth functions $f$, and bound $\langle \phi\_j, f-\chi\_A \rangle$ by $\Vert \phi\_j \Vert \Vert f-\chi\_A \Vert$. 


