Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem $$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times M,$$along with suitable initial condition.

Let $\chi\in C^\infty_c(\mathbb{R})$. For a constant $\lambda>0$ I am looking for supnorm of $$L(x,y):=\int_{\mathbb{R}}\chi(t)e^{i\lambda t}K(t,x,y)dt$$ in term of $\lambda$ where $x,y$ run over $M$.

Sogge's book "Hangzhou lectures on Eigenfunction of the Laplacian" has similar content in the chapter 3 [see section 3.6]. He used Hadamard parametrix to prove that if $\mathrm{dist}_M(x,y)=r$ [eqns (3.6.7),(3.6.8),(3.6.15)] $$|L(x,y)| \ll \begin{cases} \lambda, \text{ if }x=y\\ \sqrt{\lambda/r},\text{ if }r\ge 1 \end{cases}.$$

I am not sure how to get supnorm from here.

Equivalently, is it possible to prove that $L(x,y)$ attains supremum on $x=y$?


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