We provide some new integral estimates for solutions to Hamilton-Jacobi equations and we discuss several consequences, ranging from L p -rates of convergence for the vanishing viscosity approximation to regularizing effects for the Cauchy problem in the whole Euclidean space and Liouville-type theorems. Our approach is based on duality techniques à la Evans and a careful study of advection-diffusion equations. The optimality of the results is discussed by several examples.