# Equivalence of Gradient and Hessian Norms in Sobolev Space

I am attempting to demonstrate that for $$u \in W^{2,p}(U) \cap W^{1,p}_0(U)$$ \begin{align*} \int_U |Du|^p \ dx \leq C\int_U |D^2u|^p \ dx \end{align*}

where $$U$$ is some open subset of $$\mathbb{R}^n$$.

All that I have been able to deduce is that \begin{align*} \int_U |Du|^p \ dx \leq C \int_U |u||Du|^{p - 2} |D^2 u| \ dx \end{align*}

by using integration by parts, and I have been unsuccessful with the general Hölder's inequality to demonstrate the required inequality.