Construction of the Green function on a Riemannian manifold using harmonic coordinates (Q1816603)

Let \((M,g)\) be a compact Riemannian manifold of dimension \(n\geq 3\) without boundary. If \(\nabla\), resp. \(\Delta\), denote the Levi-Civita connection of \((M,g)\), resp. the Laplace operator, the author proves the estimate \(|\nabla^2 u|_p\leq C|\Delta u|_p\), where \(C\) depends on the geometric data of \((M,g)\). This \(L^p\)-estimate is derived via a Calderon-Zygmund type inequality. In the main result of the paper it is estimated the constant \(C\) in terms of the diameter, the injectivity radius and the lower bound of the Ricci tensor. For this aim the author constructs a parametrix of the Green function by using harmonic coordinates.