Pólya's conjecture for hyperbolic balls and Berezin--Li--Yau type inequalities

We prove the strict shifted Dirichlet Pólya inequality \[ λ_k(B_R)>(d-1)^2κ^2/4+4π^2(k/ω_dVol_κ(B_R))2/d, k≥1, \] for every geodesic ball BR in hyperbolic space of constant sectional curvature -κ², in every dimension d≥2. As a consequence, we obtain the shifted Berezin--Li--Yau type inequality for hyperbolic balls, thereby providing a concrete family of domains satisfying both conjectures proposed by Laptev in 2023. For arbitrary nonempty open subsets Ωᵈ_κ of finite volume, we establish the weak Berezin--Li--Yau type inequality \[ ∑ⱼ₌₁ᵐλ_j(Ω)≥ ad,κm+d/d+2\,4π^2(m/ω_dVol_κ(Ω))2/dm, m≥1, \] with the optimal leading coefficient, where a2,κ=κ²/4 and ad,κ=d(d-1)κ²/6 for d≥3. We also establish the global two-sided estimate for the renormalized eigenvalue sum on hyperbolic space, originally proposed by Strichartz in 1996 and valid for d≥3.