Sharp $L^q$ estimate of adjoint problem and Pucci conjecture

Let $L$ be a linear second-order elliptic operator satisfying a nondegeneracy condition. We establish sharp $L^q$ estimates for solutions of the adjoint problem associated with $L$ and, by duality, derive sharp $L^p$ estimates for solutions of $Lu=f$. In particular, this confirms a conjecture of Carlo Pucci.