Effective correlation and decorrelation for newforms, and weak subconvexity for L-functions

Let f and g be spectrally normalized holomorphic newforms of even weight k ≥2 on Γ₀(q). If f≠ g, then assume that q is squarefree. For a nice test function ψ supported on Γ₀(1), we establish the best known bounds (uniform in k, q, and ψ) for \[ ∫Γ_0(q)ψ(z)f(z)g(z)̄yᵏdxdy/y^2-1f = g3/π∫Γ_0(1)ψ(z)dx dy/y^2.\] When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When f ≠ g, our results extend and improve the effective decorrelation result of Huang for $q=1$. To prove our results, we refine Soundararajan's weak subconvexity bound for Rankin-Selberg L-functions.