Integrable functions in negative Sobolev spaces

Statement

If $f\in L^1(\mathbb R^n)$ and $s>n/2$ , then

\|f\|_{H^{-s}}\le C_{n,s}\|f\|_1.

Thus an integrable function defines an element of the negative Sobolev space $H^{-s}$ .

Proof and bounded multipliers

The Fourier integral obeys $|\widehat f(\xi)|\le\|f\|_1$ . The weight integral $\int(1+4\pi^2|\xi|^2)^{-s}\,d\xi$ is finite precisely for $2s>n$ . Substitution proves the bound. More generally, if $m$ is bounded and measurable, the inverse Fourier transform of the ordinary function $m\widehat f$ belongs to $H^{-s}$ with bound $C_{n,s}\|m\|_\infty\|f\|_1$ . This construction does not require defining multiplication of an arbitrary distribution by a nonsmooth multiplier.