ScalingStacks

Remark 6. To appreciate the strength of Kolodziej’s results, these should be contrasted with the Poisson equation $\Delta u=f$ on a compact Riemannian manifold in dimensions at least three. If $f\in L^{p}$ for some $p>1$, then we can only deduce $u\in W^{2,p}$, and $W^{2,p}$ fails to embed into $L^{\infty}$ for $p$ close to one, so we cannot expect any a priori $L^{\infty}$ bound on $u$. Ultimately, the global positivity condition $\phi\in PSH(X,\omega)$ makes the difference.