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I have managed to show that:

(a) The Dirac function $\delta_0 \notin H^{-\frac{n}{2}}(\mathbb{R^n})$, and

(b) $\eta_{\varepsilon}(x)= \varepsilon^{-n} \eta(\frac{x}{\varepsilon})$ converges to $\delta_0$ in $\mathscr{S}'$ when $\varepsilon \rightarrow 0^+$, in which $\eta \in C_0^{\infty} (\mathbb{R}^n)$ and $\int \eta =1$.

How can I show $L^1$ cannot be embedded into $H^{-\frac{n}{2}}(\mathbb{R^n})$ from (a) and (b)? This might be a one-step thing, yet I am not quite familiar with all those embedding stuffs.

Any hints or introductions of theorems to be used will be greatly appreciated. Thanks!

bvbarsenal
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1 Answers1

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The family $\{\eta_\epsilon\}_{\epsilon>0}$ is uniformly bounded in $L^1$. If $L^1$ was continuously embedded in $H^{-n/2}$ then along a sequence $\epsilon_n\to 0$, it would converge weakly in $H^{-n/2}$ to some $\eta \in H^{-n/2}$. In particular then, $\eta_{\epsilon_n}\to \eta$ in $\mathscr S'$. But this contradicts (a) when you take into account uniqueness of limits in $\mathscr S'$.

Calvin Khor
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