Proposition 16

Let $p\in [1,+\infty ]$ and $f\in \mathscr{L}^p(\mathbf{R}^n)$. Then the measure $f\cdot \mu$ with density $f$ with respect to Lebesgue measure is tempered. The mapping $f\mapsto f\cdot \mu$ of $L^p(\mathbf{R}^n)$ into $\mathscr{S}'(\mathbf{R}^n)$ resulting is continuous.

Let $q$ be the conjugate exponent of $p$ and let $r\geqslant 0$ be such that $rq > n$. For every $x\in \mathbf{R}^n$, let us write $g(x) = (1 +\|x\|)^{-r}$. The function $g$ belongs to $\mathscr{L}^q(\mathbf{R}^n)$ by Prop. 3 of IV, p. 199. By Hölder's inequality, one has

therefore the measure $f\cdot \mu$ is tempered.

Let $f\in L^p(\mathbf{R}^n)$ and $\varphi \in \mathscr{S}(\mathbf{R}^n)$. Hölder's inequality implies

and the continuity of the mapping $f\mapsto f\cdot \mu$ then follows from Prop. 13 of IV, p. 213.

For every $p\in [1,+\infty ]$, we shall identify $L^p(\mathbf{R}^n)$ with a subspace of $\mathscr{S}'(\mathbf{R}^n)$ by the linear mapping $f\mapsto f\cdot \mu$.