Local Estimates for higher order homogeneous elliptic operators

Question

For $u\in W^{2k}_2(\mathbb{R^n})$, $k\geq 1$, it is well known (see, for example, Exercise 12.9.4 in Krylov, N. "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces") that the following estimate holds:

\begin{equation} [u]_{W^{2k}_2(\mathbb{R}^n)}\leq N_1 \|\sum_{j=1}^n (D^j)^{2k} u\|_{L_2(\mathbb{R}^n)}, \end{equation}

where $[u]_{W_2^{2k}}:=\sum_{|\alpha|=2k}\|D^\alpha u \|_{L_2}$, $D^j\cdot = \cdot_{x^j}$ and $N_1$ is a constant.

My question is if there is a local version of this estimate. That is, for balls $B_r$ with radius $r$ and some constant $N_2$, is it true that

\begin{equation} [u]_{W^{2k}_2(B_R)}\leq N_2 \|\sum_{j=1}^n (D^j)^{2k} u\|_{L_2(B_{2R})} \end{equation}

(possibly with additional terms on the right hand side)? If it is, could you please point me in the direction of a reference that contains a proof of this result?

I would appreciate any comments.

Answer 1

The estimate that I can prove, and that doesn't involve crazy exponents, is: $$ [u]_{W^{2k}_2(B_1)} \leq N \left( \| Lu \|_{L^2(B_2)} + \| u\|_{L^2(B_2)}\right),\qquad \text{(1)} $$ where $L=\sum (D^j)^{2k}$. Notice that the inequality you write in the question has to fail (take for example $k=1$ and $u$ any harmonic function), so that the term $\| u\|_{L^2}$ on the RHS is somewhat necessary, though it can be weakened.

In what follows $\nabla^j$ denotes the collection (tensor) of all derivatives of order $j$.

1. Consider $\eta \in C_c^\infty(\mathbb{R}^n)$, $0\leq \eta\leq 1$, such that $\eta \equiv 1$ in $B_1$ and $\text{supp} \eta \subset B_2$. Define $v=\eta u$. By the inequality on the half space we get that $$ [v]_{W^{2k}_2(\mathbb{R}^n)} \leq N \| Lv \|_{L^2(\mathbb{R}^n)}.\qquad \text{(2)} $$

2. By the choice of $\eta$ we have $\eta \equiv 1$ and $\nabla^j \eta \equiv 0$, for $1\leq j$, in $B_1$ so that, expanding the LHS of (2) we get that $$ [u]_{W^{2k}_2(B_1)} \leq C[v]_{W^{2k}_2(\mathbb{R}^n)}, \qquad \text{(3)} $$ where $C$ depends only on $\eta, n, k$.

3. To handle the RHS of (2), notice that the Gagliardo-Nirenberg interpolation inequality gives that any derivative, $1\leq j<2k$, and any $\varepsilon>0$, $$ \| \nabla^j v\|_{L^2(\mathbb{R}^n)} \leq \varepsilon [v]_{W^{2k}_2(\mathbb{R}^n)} + C_\varepsilon\| v\|_{L^2(\mathbb{R}^n)} , $$ so that any term involving a derivative of order $1\leq j<2k$ of $u$ appearing on the RHS of (2) can be bounded by $\varepsilon\| L v\|_{L^2(\mathbb{R}^n)}$. Thus, we obtain $$ \| L v\|_{L^2(\mathbb{R}^n)} \leq C_0\| L u\|_{L^2(B_2)} + C_1 \varepsilon \| Lv\|_{L^2(\mathbb{R}^n)} + C_3 \| v\|_{L^2(\mathbb{R}^n)}. $$ Choosing $\varepsilon$ so that $C_1 \varepsilon<1/2$ we arrive at $$ \| Lv\|_{L^2(\mathbb{R}^n)} \leq C\left( \| Lu\|_{L^2(B_2)} + \| u\|_{L^2(B_2)}\right). \qquad \text{(4)} $$ Plugging (3) and (4) into (2) we get the desired inequality.