Estimates of automorphic forms on $\mathrm{SU}(n,1)$

Abstract: For $n\geq 2$, let $\Gamma\subset \mathrm{SU}((n,1),\mathcal{O}{K})$ be a torsion-free, finite-index subgroup, where $\mathcal{O}_K$ denotes the ring of integers of a totally imaginary number field $K$ of degree $2$. Let $\mathbb{B}^n$ denote the $n$-dimensional complex ball endowed with the hyperbolic metric, and let $X{\Gamma}:=\Gamma\backslash \mathbb{B}^n$ denote the quotient space. Furthermore, let $\mu_{\mathrm{hyp}}^{\mathrm{vol}}$ denote the volume form associated to the hyperbolic metric. Let $\Lambda:=\Omega_{\overline{X}{\Gamma}}^{n}$ denote the line bundle, where $\overline{X}{\Gamma}:=X_{\Gamma}\cup\lbrace \infty\rbrace$. For any $k\geq 1$, let $\lambda^{k}:=\Lambda^{\otimes k}\otimes O_{\overline{X}{\Gamma}}((k-1)\infty)$. For any $k\geq 1$, the hyperbolic metric induces a point-wise metric on $H^{0}(\overline{X}{\Gamma},\lambda^{k})$. For any $k\geq 1$, let $\mathcal{B}{X{\Gamma}}^{\lambda^{k}}$ denote the Bergman kernel associated $H^{0}(\overline{X}_{\Gamma},\lambda^{k})$. Then, for $k\gg1$, the first main result of the article, is the following estimate $$ \sup_{z\in \overline{X}{\Gamma}}\big|\mathcal{B}{X_{\Gamma}}^{\lambda^{k}}(z,z)\big|{\mathrm{hyp}}=O{X_{\Gamma}}(k^{n+1/2}), $$ where the implied constant depends only on $X_{\Gamma}$. For any $k\geq 1$, and $z\in X_{\Gamma}$, let $\mu_{\mathrm{Ber},k}(z)$ denote the Bergman metric associated to the line bundle $\lambda^{ k}$, and let $\mu_{\mathrm{ber},k}^{\mathrm{vol}}$ denote the associated volume form. Then, for $k\gg1$, the second main result of the article is the following estimate $$ \sup_{z\in \overline{X}{\Gamma}}\bigg|\frac{\mu{\mathrm{Ber},k}^{\mathrm{vol}}(z)}{\mu_{\mathrm{hyp}}^{\mathrm{vol}}(z)}\bigg|=O_{X_{\Gamma}}\big(k^{4n-1} \big), $$ where the implied constant depends only on $X_{\Gamma}$.