Counterexamples to the $L^p$-Calder\'{o}n--Zygmund Estimate on Open Manifolds

Based on a construction due to B. G"{u}neysu and S. Pigola (\textit{Adv. Math.} $\mathbf{\text{281}}$ (2015), pp.353–393), for each $p\in [1,\mathrm{\infty }]$ and $m \in $\mathbb{Z}${\geq 2}$,weexhibitan$m$-dimensional Riemannian open manifold $\mathcal{M}$ on which the $L^p$-Calder'{o}n–Zygmund estimate \begin{equation*} |\nabla \nabla f|^p{L^p} \leq C_1 |\Delta f|{L^p}^p + C_2 | f|{L^p}^p \qquad $\text{ for all }$ f \in C^\infty_c($\mathcal{M}$) \end{equation*} is false for any $C_1,C_2$ depending on $m$ and $p$. Therefore, one must impose further geometric conditions on the manifold to ensure the validity of the Calder'{o}n–Zygmund estimate.