The sharp reverse Hardy inequality in BMO for nonincreasing functions

What the math gives to ML

The paper proves a sharp lower bound for the Hardy prefix-averaging operator on the cone of nonincreasing functions: averaging cannot reduce the BMO seminorm by more than the optimal factor alpha_0, and one-jump functions are extremizers. The transferable asset is not generic averaging, but a certified anti-collapse property obtained by combining prefix integration with a monotonicity constraint. This suggests a causal neural module that applies cumulative averaging to monotone gates or feature profiles while monitoring a discrete BMO ratio. The construction could provide a cheap alternative to dense causal attention when importance profiles are expected to decay with sequence position.