The Weyl functional on 4-manifolds of positive Yamabe invariant

It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $$int_M|W^+_g|^2dmu_{g}geq 2pi^2(2chi(M)+3tau(M))-frac{8pi^2}{|pi_1(M)|},$$ where $W^+_g$, $chi(M)$ and $tau(M)$ respectively denote the self-dual Weyl tensor of $g$, the Euler characteristic and the signature of $M$. This generalizes Gurskys inequality cite{gur} for the case of $b_1(M)>0$ in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gurskys inequalities for the case of $b_2^+(M)>0$ or $delta_gW^+_g=0$, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.