Weighted Hardy Inequalities on Time Scales in the Range 0 < q < 1 < p < ∞

Let 0<q<1<p<∞ and let T be an arbitrary time scale. We establish a necessary-and-sufficient two-weight criterion in the quasi-Banach range for non-negative measurable functions from Lp([a,∞)T,νp Δt) to Lq([a,∞)T,ωq Δt). The criterion is the finiteness of a mixed head–tail quantity involving U(t)=∫t∞ω(τ)q Δτ and V(σ(t))=∫aσ(t)ν(τ)−p′ Δτ, and explicit two-sided bounds are obtained for the optimal constant. The key technical ingredient is a gap-compatible weighted level-function construction based on interval averages over right-scattered gaps. The result recovers the continuous characterization of Sinnamon and the discrete characterization of Braverman and Stepanov, and it also yields dynamic averaging, Bennett–Copson-type, Hardy–Flett-type, and shifted quantum specializations.