Quantitative C^1 - estimates on manifolds

Abstract

We prove a ${\mathsf{C}}^1$-elliptic estimate of the form ${\sup }_{B(x,r/2)}|\mathrm{grad}(\psi )|\le C\left\{{\sup }_{B(x,r)}|\Delta \psi |+{\sup }_{B(x,r)}|\psi |\right\},$ valid on any complete Riemannian manifold $M$ and for any smooth function $\psi$ which is defined in a nighbourhood of $B(x,r)$, with an explicit quantitative control on the constant $C=C(B(x,r))$ in terms of the curvature of the geodesic ball $B(x,r)\subset M$.

arXiv commentan application concerning zero means of Laplacians are added; to appear in IMRN