Sobolev inequality in negative index

Question

For $s>n/2$, is it true that $$ \int |fg| dx\leq ||f||_{H^s}||g||_{H^{-s}}?$$ This inequality is used on pg 398 of the Majda Bertozzi book on Vorticity and Incompressible flow but I can't make sense of it. Thanks.

Answer 1

$H^{-s}$ is by definition the dual of $H^s$. I suppose that $\int f\,g\,dx$ must be understood in the sense of duality. Elements of $H^{-s}$ are not functions, but distributions, so that $f\,g$ is not defined in general. If for instance both $f$ and $g$ are in the Schwartz class, then $$ \Bigl|\int f\,g\,dx\Bigr|=\Bigl|\int\hat f\,\hat g\,d\xi\Bigr|=\Bigl|\int\hat f\,\langle\xi\rangle^{s/2}\,\hat g\,\langle\xi\rangle^{-s/2}\,d\xi\Bigr|\le\|f\|_{H^s}\,\|g\|_{H^{-s}}. $$