Determinacy of refinements to the difference hierarchy of co-analytic sets

In this paper we develop a technique for proving determinacy of classes of the form ω2-Π11+Γ (a refinement of the difference hierarchy on Π11 lying between ω2-Π11 and (ω2+1)-Π11) from weak principles, establishing upper bounds for the determinacy-strength of the classes ω2-Π11+Σα0 for all computable α and of ω2-Π11+Δ11. This bridges the gap between previously known hypotheses implying determinacy in this region.