High-frequency control by a Sobolev derivative
= High-frequency control by a Sobolev derivative
For $u\in H^1(\mathbb R^d)$, Plancherel gives
$$
\int_{|\xi|>R}|\widehat u(\xi)|^2\,d\xi
\leq R^{-2}\lVert\nabla u\rVert_2^2.
$$
= High-frequency control by a Sobolev derivative
For $u\in H^1(\mathbb R^d)$, Plancherel gives
$$
\int_{|\xi|>R}|\widehat u(\xi)|^2\,d\xi
\leq R^{-2}\lVert\nabla u\rVert_2^2.
$$