A Quantitative Dolgopyat Estimate on Exponentially Weighted Banach Spaces Bθ,σ,α

We prove a quantitative Dolgopyat-type bound for the Gauss-Mayer transfer operator acting on the exponentially weighted Banach space B_{θ,σ,α}:

||A_{σ+it}||{B{θ,σ,α}→B_{θ,σ,α}} ≤ C(θ,α,ε) |t|^{-1/4}, |t| ≥ 1, |σ - 1/2| ≥ ε > 0.

The constant C(θ,α,ε) is explicit and depends polynomially on ε^{-1}. The exponential decay parameter α > 0 is essential for the bound to hold—no polynomial decay in |t| exists when α = 0. Our argument combines a uniform stationary-phase estimate with a weighted Schur test to achieve the optimal |t|^{-1/4} decay rate.