A reciprocal phase lemma and sharp rates for detuned triangular semigroups

For every fixed alpha > 1, gamma >= 0 and 1 <= p < infinity, we study a triangular semigroup on l^p(N;C^2) with integer frequencies and the original n^(-1) detuning. We prove sharp two-sided growth of order (1+t)^((1-gamma)_+) for every real t >= 0, and regularized decay of order t^(-min(gamma,1)/alpha) for every real t >= 1. An integer reciprocal phase lemma supplies the matching lower bound when alpha > 2.