In this article, we give \(L^{\infty }\) -norm bounds for the natural invariant norm of cusp forms of real weight k and character \(\chi \) for any cofinite Fuchsian subgroup \(\Gamma \subset \textrm{SL}_{2}(\mathbb {R})\) . Using the representation of Jacobi cusp forms of integral weight k and index m for the modular group \(\Gamma _{0}=\textrm{SL}_{2}(\mathbb {Z})\) as linear combinations of modular forms of weight \(k-\frac{1}{2}\) for some congruence subgroup of \(\Gamma _{0}\) (depending on m) and suitable Jacobi theta functions, we derive \(L^{\infty }\) -norm bounds for the natural invariant norm of these Jacobi cusp forms. More specifically, letting \(J_{k,m}^{\textrm{cusp}}(\Gamma _{0})\) denote the complex vector space of Jacobi cusp forms under consideration and \(\Vert \cdot \Vert _{\textrm{Pet}}\) the pointwise Petersson norm on \(J_{k,m}^{\textrm{cusp}}(\Gamma _ {0})\) , we prove that for \(k\in \mathbb {Z}_{\ge 5}\) and \(m\in \mathbb {Z}_{\ge 1}\) , and a given \(\epsilon >0\) , the \(L^{\infty }\) -norm bound \(\begin{aligned} \Vert \phi \Vert _{L^{\infty }}=\sup _{(\tau ,z)\in \mathbb {H}\times \mathbb {C}}\Vert \phi (\tau ,z)\Vert _{\textrm{Pet}}=O_{\Gamma _{0},\epsilon }\big (k\,m^{\frac{7}{4}+\epsilon }\big ) \end{aligned}\) holds for any \(\phi \in J_{k,m}^{\textrm{cusp}}(\Gamma _{0})\) , which is \(L^{2}\) -normalized with respect to the Petersson inner product, where the implied constant depends on \(\Gamma _{0}\) and the choice of \(\epsilon >0\) .
