Stefano Pigola

I will give an overview of some recent results concerning the validity and the failure of a global estimate of the form  $ | \nabla u |_{L^p(M)} + | \mathrm{Hess}(u) |_{L^p(M)} \leq C { | u |_{L^p(M)} + | f |_{L^p(M)}}$, where $u$ is a smooth solution of the Poisson equation $\Delta u = f$ and $C>0$ is a constant that depends on the geometry of the underlying complete Riemannian manifold $(M,g)$.