A?-weights and compactness of conformal metrics under Ln?2 curvature bounds

We study sequences of conformal deformations a smooth closed Riemannian manifold dimension $n$, assuming uniform volume bounds and $L^{n/2}$ on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such underlying metric spaces are pre-compact Gromov-Hausdorff topology. Our is based use $A_\infty$-weights from harmonic analysis, provides geometric controls limit thus obtained. techniques also any deformation Euclidean $R^n$ with infinite finite norm curvature satisfies isoperimetric inequality.