= Constant-coefficient elliptic second-derivative estimate
{title2=$\theta\|D^2u\|_2\leq\|A^{ij}D_{ij}u\|_2$}
If the constant <symmetric matrix> $A$ satisfies $A\xi\mathbin\cdot\xi\geq\theta|\xi|^2$, then every <smooth function> $u$ with <compact support> satisfies
$$
\theta\|D^2u\|_2\leq\|A^{ij}D_{ij}u\|_2.
$$
Indeed, the <Fourier transform of a derivative> and the <Plancherel theorem> turn the squared norms into integrals with <Fourier multipliers> $|\xi|^4$ and $(A\xi\mathbin\cdot\xi)^2$.
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