The paper gives new bounds for Maass forms of odd level \(N\) on \(PGL_1(D)\) where \(D\) is a quaternion algebra ramified away from prime divisors of \(N\). The assumption also includes that \(D\) splits at infinity and the Maass form is spherical at infinity.
The first result is a bound on \(L^{\infty}\) norm of Maass form with respect to \(N\) and spectral parameter of the form. Worth noting is that this bound in level aspect is achieved without assumption \(N\) is square-free.
The other result is the bound of \(L^p\) norm with further assumption that \(N=q^2\) with \(q\) an odd prime and the Maass form is a principal series at prime \(q\).
The proof uses harmonic analysis and trace formula.