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How to prove that functions in $L^p(\mathbb{R}^n),1 \leq p \leq \infty$, are tempered distributions.

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If $f\in L^p(\mathbb R^n)$ and $\varphi\in\mathscr S(\mathbb R^n)$, then $$ \ell(\varphi)=\int_{\mathbb R^n}f\,\varphi, $$ is definable, since $f\varphi\in L^1(\mathbb R^n)$, as $\varphi\in L^q(\mathbb R^n)$, for all $p$, and $$ \lvert \ell(\varphi)\rvert\le\|\varphi\|_{p'}\|\,f\|_p, $$ where $\frac{1}{p}+\frac{1}{p'}=1$.

All the other properties are straight-forward.

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  • $\begingroup$ straight-forward part ? $\endgroup$ – starry1990 Oct 16 '14 at 16:09
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Let $f\in L^p(\mathbb{R}^n)$. The canonical distribution corresponding to $f$ is defined by $d_f:\mathcal{S}(\mathbb{R}^n)\rightarrow \mathbb{C}$, $g\mapsto \int_{\mathbb{R}^n} fg$.

Now check that $d_f$ is an element of $\mathcal{S}^\prime(\mathbb{R}^n)$. It is clear that it is a linear map. So it remains to check continuity. Namely, we have to check that

$$\|d_f\|_{\mathcal{S}^\prime(\mathbb{R}^n)}=\sup_{\|g\|_{\mathcal{S}}=1} \left|\int_{\mathbb{R}^n} fg\right|<\infty$$

But let $g$ be a Schwartz function of Schwartz norm $1$. Then it is in all $L^p$ spaces, in particular in $L^q$ with $1/p+1/q=1$, so the claim follows by Hölder's inequality:

$$\int_{\mathbb{R}^n} |fg| \le \|f\|_p \|g\|_q<\infty$$

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