This is an exercise (10.3.4 in the third edition) from Nicolaescu's Lectures on the Geometry of Manifolds. Let $L$ be an elliptic differential operator of order $k$ and $1 < p < \infty$. The book proves that for every $R > 0$ there exists a constant $C$ depending on $L, p, R$ such that \begin{equation} \|u\|_{W^{k,p}} \leq C \left(\|L u\|_{L^p} + \|u\|_{L^p}\right) \label{eq:cpt}\tag1 \end{equation} for any $u \in C^\infty_c(B_R)$.

The exrcise consists of using this to prove that for any $0 < r < R$ there exists a constant $K$ depending on $L, p, r, R$ such that $$ \|u\|_{W^{k,p}(B_r)} \leq K \left(\|L u\|_{L^p(B_R)} + \|u\|_{L^p}(B_R)\right) $$

My approach: The book says to pick a cutoff and use interpolation inequalities and I can almost make this work.

I pick a smooth $\eta$ that is $\equiv 1$ on $B_r$ and $\equiv 0$ outside of $B_R$ and apply (\ref{eq:cpt}). This gives \begin{equation} \|u\|_{W^{k,p}(B_r)} \leq \|\eta u\|_{W^{k,p}} \leq C\left(\|L \eta u\|_{L^p} + \|\eta u\|_{L^p}\right) \leq C\left(\|L u\|_{L^p(B_R)} + \|[L, \eta] u\|_{L^p} + \|u\|_{L^p(B_R)}\right) \tag2\label{eq:int} \end{equation} The problem therefore reduces to controlling $\|[L, \eta] u\|_{L^p}$. This is bounded by $C' \|u\|_{W^{k-1,p}(B_R \setminus B_r)}$ or even by $\|u\|_{W^{k-1,p}(B_r' \setminus B_r)}$ for some $r' \in (r, R)$ if we pick $\eta$ to be supported in $B_{r'}$. Controlling this should be easy because we have the interpolation inequality $$\renewcommand{\epsilon}{\varepsilon} \|u\|_{W^{k-1,p}(B_{r'} \setminus B_r)} \leq C'' \left( \epsilon \|u\|_{W^{k,p}(B_{r'} \setminus B_r)} + \epsilon^{1-k} \|u \|_{L^p(B_{r'} \setminus B_r)}\right) $$ I would like to absorb the $W^{k,p}$ norm in the LHS of (\ref{eq:int}) but of course that doesn't work because the norm is taken over a different set.

This is an exercise (10.3.4 in the third edition) from Nicolaescu's Lectures on the Geometry of Manifolds. Let $L$ be an elliptic differential operator of order $k$ and $1 < p < \infty$. The book proves that for every $R > 0$ there exists a constant $C$ depending on $L, p, R$ such that \begin{equation} \|u\|_{W^{k,p}} \leq C \left(\|L u\|_{L^p} + \|u\|_{L^p}\right) \label{eq:cpt}\tag1 \end{equation} for any $u \in C^\infty_c(B_R)$.

The exrcise consists of using this to prove that for any $0 < r < R$ there exists a constant $K$ depending on $L, p, r, R$ such that $$ \|u\|_{W^{k,p}(B_r)} \leq K \left(\|L u\|_{L^p(B_R)} + \|u\|_{L^p}(B_R)\right) \tag2\label{eq:goal} $$

My approach: The book says to pick a cutoff and use interpolation inequalities and I can almost make this work.

I pick a smooth $\eta$ that is $\equiv 1$ on $B_r$ and $\equiv 0$ outside of $B_R$ and apply (\ref{eq:cpt}). This gives \begin{equation} \|u\|_{W^{k,p}(B_r)} \leq \|\eta u\|_{W^{k,p}} \leq C\left(\|L \eta u\|_{L^p} + \|\eta u\|_{L^p}\right) \leq C\left(\|L u\|_{L^p(B_R)} + \|[L, \eta] u\|_{L^p} + \|u\|_{L^p(B_R)}\right) \tag3\label{eq:int} \end{equation} The problem therefore reduces to controlling $\|[L, \eta] u\|_{L^p}$. This is bounded by $C' \|u\|_{W^{k-1,p}(B_R \setminus B_r)}$ or even by $\|u\|_{W^{k-1,p}(B_r' \setminus B_r)}$ for some $r' \in (r, R)$ if we pick $\eta$ to be supported in $B_{r'}$. Controlling this should be easy because we have the interpolation inequality $$\renewcommand{\epsilon}{\varepsilon} \|u\|_{W^{k-1,p}(B_{r'} \setminus B_r)} \leq C'' \left( \epsilon \|u\|_{W^{k,p}(B_{r'} \setminus B_r)} + \epsilon^{1-k} \|u \|_{L^p(B_{r'} \setminus B_r)}\right) $$ I would like to absorb the $W^{k,p}$ norm in the LHS of (\ref{eq:int}) but of course that doesn't work because the norm is taken over a different set.

The idea of using $r'$ instead of $R$ is that now the problem has been reduced to showing (\ref{eq:goal}) holds with the $W^{k-1,p}$ norm on the LHS,