TITLE: Mean value of Maass forms
QUESTION [7 upvotes]: 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.

REPLY [9 votes]: 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 $\Vert \phi_j \Vert \Vert f-\chi_A \Vert$.