Polynomial Sobolev embedding into L∞ for general multiplicative representations

Develop an L∞ bound with polynomial Sobolev control for arbitrary multiplicative representations of PSL2(R) with discrete spectrum: prove the existence of s ≥ 0 such that ∥α∥_{L∞} ≤ ∥(Δ+1)^s α∥_{H} for all α ∈ H^K ∩ H^{fin}, without relying on the classification obtained in Theorem 1.1.

Background

The authors establish a quasi-Sobolev embedding into L∞ with quasipolynomial dependence in general (Theorem 4.5), and show that for H = L^2(Γ \ G) one can strengthen this to polynomial dependence using elliptic regularity on compact manifolds.

They explicitly state that they do not know how to obtain the polynomial-strength L∞ bound for general multiplicative representations without invoking the classification (Theorem 1.1), highlighting a concrete analytic gap independent of the classification theorem.