My question here relates to an argument that I have seen used many a time, for example to prove that $\mathbb{L}^p$ spaces define tempered distributions in the following sense : $\varphi \mapsto \int \varphi f$ is a tempered distribution for $f \in \mathbb{L}^p$
The argument is, if $f \in \mathbb{L}^p$ then for any function $\varphi \in \mathbb{L}^q$ (hence for any Schwartz function)
$$
|\varphi (x)|=(1+|x|)^{N+1} \cdot |\varphi (x)| \frac{1}{(1+|x|)^{N+1}}
$$
where $N$ is the dimension of the euclidean space, and $|\cdot |$ denotes the euclidean norm. Then, $\frac{1}{(1+|x|)^{N+1}}$ is in $\mathbb{L}^q$ and one can write
$$ ||\varphi (x)||_q \leq \sup|(1+|x|)^{N+1} \varphi (x)| \cdot ||\frac{1}{(1+|x|)^{N+1}}||_q $$
What I don't get is how this $\sup$ expression can be upper-bounded by any of the Schwartz norms since $(1+|x|)^{N+1}$ may well not be a polynomial of $x$ with whole powers because of the square root.
Especially confused as I've just worked out that $|x|^{|\alpha|} \geq |x^\alpha|$ ?
Best Answer
To show that $f \in L^p(\mathbb{R}^n)$ is in fact a tempered distribution, we need to verify whether $f$ acting on Schwatz class test functions produces a finite number, i.e. for $\phi \in \mathcal{S}(\mathbb{R}^n)$ \begin{equation} |\langle f,\phi \rangle | < +\infty \end{equation} To see this, observe that by Schwartz inequality,
$$\begin{eqnarray} |\langle f,\phi \rangle |&=& \bigg|\int f \phi \bigg| \nonumber \\ &\leq& \int \big|f \phi \big| \nonumber \\ &\le& \bigg(\int \big|f \big|^p\bigg)^{1/p} \bigg(\int \big| \phi \big|^q\bigg)^{1/q} \text{ where } \frac{1}{p} + \frac{1}{q}=1\\ &=& C_f || \phi||_{L^q} \text{ where } C_f=||f||_{L^p}\\ &<& +\infty \text{ as } \phi \in L^q \end{eqnarray}$$