Weighted Sup-Norms at Infinity: Exhaustion Functions, Decay Classes, and Anisotropic Ends
For a non-compact, locally compact, sigma-compact Hausdorff space with a continuous proper exhaustion function, and an admissible weight, we study the weighted supremum norms measuring how fast a function approaches its limit at infinity, and their quotient modulo constants. Our main result is a reduction theorem: when the weight is unbounded, the infimum over constants is attained at the limit at infinity, and multiplication by the weight is a surjective linear isometry onto the bounded continuous functions, carrying the decay class onto the functions vanishing at infinity. Completeness, duality, compactness and density of truncations are therefore classical facts read through this isometry. We then correct two natural expectations. The weighted zero-mass measures embed isometrically into the dual of the decay class with dense range, but are not the whole dual; norming measures exist exactly when the transformed function attains both its supremum and its negative, which fails generically. And proper exhaustions are not in general coarsely affinely equivalent, so the decay classification depends on the exhaustion; we characterise coarse-affine invariance by a dilation condition on the weight along the image of the exhaustion. For spaces with finitely many ends, the single-constant quotient is infinite as soon as two ends carry different limits; we replace it by an end-by-end functional admitting a structure theorem and completeness. An appendix lists the corrections made with respect to the posted version.