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 $\omega^2-\Pi^1_1+\Gamma$ (a refinement of the difference hierarchy on the co-analytic sets lying between $\omega^2-\Pi^1_1$ and $(\omega^2+1)-\Pi^1_1$) from weak principles, establishing upper bounds for the determinacy-strength of the classes $\omega^2-\Pi^1_1+\Sigma^0_\alpha$ for all computable $\alpha$ and of $\omega^2-\Pi^1_1+\Delta^1_1$. This bridges the gap between previously known hypotheses implying determinacy in this region.