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