This is a problem from Stein, Shakarchi, Real Analysis, Chapter 5, Exercise 18. We let $L$ be a constant-coefficient partial differential operator $$L = \sum_{|\alpha|\leq n} a_\alpha (\partial_x^\alpha) $$ and let $P$ denote its "characteristic polynomial" $$P(\xi) = \sum_{|\alpha|\leq n} a_\alpha (2\pi i \xi)^\alpha.$$ We define $L$ to be elliptic iff $|P(\xi)|\geq |\xi|^n$ for all large $\xi$, or iff $\sum_{|\alpha|=n} a_\alpha (2\pi i \xi)^\alpha$ is non-vanishing for $\xi\neq 0$.
Here's the desired estimate: show that for any test function $u\in C_c^\infty(\mathbb{R}^d)$ and any multi index $\alpha$ with $|\alpha|\leq n$, we have $$\|\partial_x^\alpha u\|_{L^2(\mathbb{R}^d)} \lesssim \|Lu\|_{L^2(\mathbb{R}^d)} + \|u\|_{L^2(\mathbb{R}^d)}.$$ I believe that the most obvious? way to approach is with the Fourier transform. Clearly $$\|\partial_x^\alpha u\|_2^2 = \int_{\mathbb{R}^d} (2\pi i \xi)^\alpha |\hat{u}(\xi)|^2 d\xi $$ and $$\|Lu\|_2^2 = \int_{\mathbb{R}^d} \left|\sum_{|\alpha|\leq n} a_\alpha (2\pi i \xi)^\alpha \right|^2 |\hat{u}(\xi)|^2 d\xi = \int_{\mathbb{R}^d} |P(\xi)|^2 |\hat{u}(\xi)|^2.$$ After this step, I'm basically stumped... I would appreciate all help.
P.S. In this book we define the Fourier transform of a Schwarz function as $$\hat{u}(\xi) = \int u(x) e^{-2\pi i x \cdot \xi} dx.$$
Note that we can bound below by restricting to where the elliptic bound holds, and so $$\|Lu\|_2^2\gtrsim \int_{|\xi|\gg 1}|\xi|^{2n}|\hat{u}(\xi)|^2\, d\xi\gtrsim \int_{|\xi|\gg 1} \xi^{2\alpha}|\hat{u}(\xi)|^2\, d\xi$$ for any $|\alpha|\leq n.$ Adding back in the missing contribution, $$\|\partial^\alpha u\|_2^2\lesssim \|Lu\|_2^2+\int_{\mathbb{R}^d\setminus \{|\xi\| \gg 1\}}\xi^{2\alpha}|\hat{u}(\xi)|^2\lesssim \|Lu\|_2^2+\|u\|_2^2,$$ since the integration region in the last term is compact.