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The spectrum of the Laplacian on $L^2$ of a Hilbert modular surface decomposes into a discrete part and a continuous part $[1/4,\infty)$. The continuous part contains eigenvalues $\geq 1/4$.

I would like to know if the eigenfunctions of such eigenvalues lie also in $L^p$ for some $p>2$. If so, why?

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    $\begingroup$ For dimension one the answer is yes, as all those eigenfunctions are cusp forms which decay exponentially at the cusps. I think a similar argument should settle the case of Hilbert forms as well. $\endgroup$
    – user1688
    Commented Jan 11, 2017 at 11:37

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