Estimates of automorphic forms on SU(n,1)

For n≥ 2, let Γ⊂ SU((n,1),OK) be a torsion-free, finite-index subgroup, where OK denotes the ring of integers of a totally imaginary number field K of degree $2$. Let Bⁿ denote the n-dimensional complex ball endowed with the hyperbolic metric, and let XΓ:=Γ Bⁿ denote the quotient space. Furthermore, let μhypᵛᵒˡ denote the volume form associated to the hyperbolic metric. Let Λ:=Ω_X̄Γⁿ denote the line bundle, where X̄Γ:=XΓ∪ ∞. For any k≥ 1, let λᵏ:=Λ⊗ k⊗ O_X̄Γ((k-1)∞). For any k≥ 1, the hyperbolic metric induces a point-wise metric on H⁰(X̄Γ,λᵏ). For any k≥ 1, let B_XΓ^λᵏ denote the Bergman kernel associated H⁰(X̄Γ,λᵏ). Then, for k1, the first main result of the article, is the following estimate _z∈ X̄Γ|B_XΓ^λᵏ(z,z)|hyp=O_XΓ(kn+1/2), where the implied constant depends only on XΓ. For any k≥ 1, and z∈ XΓ, let μBer,k(z) denote the Bergman metric associated to the line bundle λᵏ, and let μber,kᵛᵒˡ denote the associated volume form. Then, for k1, the second main result of the article is the following estimate _z∈ X̄Γ|μBer,kᵛᵒˡ(z)μhypᵛᵒˡ(z)|=O_XΓ(k⁴ⁿ⁻¹ ), where the implied constant depends only on XΓ.