Stability estimate for the Lane-Emden inequality

Abstract

The Lane-Emden inequality controls ${\iint }_{{\mathbb{R}}^{2d}}\ddot{I}(x)\ddot{I}(y)|x-y{|}^{-\hat{I}»}dxdy$ in terms of the $L^1$ and $L^p$ norms of $\ddot{I}$. We provide a remainder estimate for this inequality in terms of a suitable distance of $\ddot{I}$ to the manifold of optimizers.