About Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec

Question

In Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec, the author said that the kernel of the invariant integral operator $$ (Lf)(z)=\int_{\mathbb{H}}k(z,w)f(w)d\mu w $$ is not bounded on $F\times F$. To avoid this difficulty he used the so called principal parts $$ H_{a}(z,w)=\sum_{\gamma\in\Gamma_{a}\backslash\Gamma}\int_{-\infty}^{+\infty}k(z,\sigma _{a}n(t)\sigma_{a}^{-1}\gamma w) dt, $$ and showed that the kernel $$ K(z,w)-\sum_{a}H_{a}(z,w) $$ is bounded. Finally, he deduced that the operator of $K(z,w)-\sum_{a}H_{a}(z,w)$ is of Hilbert-Schmidt type.

I didn't demystify yet the proof of Proposition 4.5, but in Hejhal's book (The Selberg Trace Formula for $PSL(2,\mathbb{R})$, page 14) the same operator $L$ is considered to be of Hilbert-Schmidt type and thus the kernel $K(,)$ must be bounded!!

I need to know which of the two books is telling the truth?

Thank you for your help.

Answer 1

Hejhal uses that $K\in L^2(F\times F)$, which follows from the continuity of $K$ and the compactness of $F$. The latter assumption appears on the first page of Hejhal's book: "Let $F$ denote a compact Riemann surface of genus $g\geq 2$." Iwaniec deals with the general case. So both Hejhal and Iwaniec are right, but Iwaniec is more general.