The $L^p$-Calderón–Zygmund inequality on non-compact manifolds of positive curvature

Abstract

We construct, for $p>n$, a concrete example of a complete non-compact n-dimensional Riemannian manifold of positive sectional curvature which does not support any $L^p$-Calderón–Zygmund inequality:

$$\begin{aligned}\Vert {{\,\mathrm{Hess}\,}}\varphi \Vert _{L^p}\le C(\Vert \varphi \Vert _{L^p}+\Vert \Delta \varphi \Vert _{L^p}), \qquad \forall \,\varphi \in C^{\infty }_c(M). \end{aligned}$$

The proof proceeds by local deformations of an initial metric which (locally) Gromov–Hausdorff converge to an Alexandrov space. In particular, we develop on some recent interesting ideas by De Philippis and Núñez–Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the $L^p$-gradient estimates and the $L^p$-Calderón–Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.