In this article [Interpolation inequalities with weights
Chang Shou Lin](https://www.tandfonline.com/doi/abs/10.1080/03605308608820473?journalCode=lpde20) the following lemma is stated and proved.


Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$ such that


$$|||x|^{\alpha}u||\_p \leq C |||x|^{\alpha+1}Du||\_p.$$


the proof consists only of the direct calculation
\begin{align}
|||x|^{\alpha}u||\_p^p& = \int|x|^{p\alpha}|u|^p dx\\
& \leq C \int|x|^{p\alpha+1}|u|^{p-1}|Du| dx\\
& \leq C \left( \int|x|^{p\alpha}|u|^p dx \right)^{1-1/p} \left( \int|x|^{p(\alpha+1)}|Du|^p dx \right)^{1/p}
\end{align}
I'm almost certain that in the first inequality, we just used integration by parts. Hence, it is necessary to assume that the function u has compact support (the lemma does not make it clear what the function's class is), I got
$$-(p\alpha+1)\int |x|^{p\alpha} \dfrac{\vec{x}}{|x|} u^pdx = p \int |x|^{p\alpha+1}u^{p-1}Dudx $$
I guess I must be missing something because I'm not using the hypothesis $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$. Or maybe that's not how it's done. Does anyone have any suggestions? Also, I'm thinking that the proper space in which this inequality holds is the space of C^1 functions with compact support, is that the best space? The last inequality is Holder's inequality.



Lemma about the weighted interpolation inequality
$\newcommand\R{\mathbb R}\newcommand{\al}{\alpha}$In view of conditions (0.1) and (0.4) in the linked paper, $p\in[1,\infty)$ and $u\in C\_0^\infty(\R^n)$. To prove the lemma in question (Lemma 2.2 in that paper), it suffices that $u\in C\_0^1(\R^n)$.


Indeed,
\begin{equation\*}
 \|\,|x|^\al u\|\_p^p=I:=\int\_{\R^n}dx\,|x|^{p\al} |u(x)|^p. \tag{5}\label{5}
\end{equation\*}
In view of the condition $u\in C\_0^1(\R^n)$,  

\begin{equation\*}
 -|u(x)|^p=\int\_1^\infty dt\,pu(tx)^{[p-1]}(Du)(tx)\cdot x,
\end{equation\*}
where $z^{[p-1]}:=|z|^{p-2}z$ for real $z\ne0$, with $0^{[p-1]}:=0$, $(Du)(y)$ is the gradient of $u$ at $y$, and $\cdot$ denotes the dot product.
So,  

\begin{equation\*} 
\begin{aligned} 
I&\le\int\_{\R^n}dx\,|x|^{p\al+1} \int\_1^\infty dt\,p|u(tx)|^{p-1}|Du|(tx) \\ 
&=C\int\_{\R^n}dy\,|y|^{p\al+1} \,p|u(y)|^{p-1}|Du|(y), 
\end{aligned}
\end{equation\*}
where
\begin{equation\*}
 C:=\int\_1^\infty \frac{dt}{t^{n+p\al+1}}=\frac1{n+p\al},
\end{equation\*}
given the condition
\begin{equation\*}
 \frac1p+\frac\al n>0, \tag{10}\label{10}
\end{equation\*}
which can be rewritten as $n+p\al>0$.


So, using Jensen's inequality as in your post, we get
\begin{equation\*}
 I\le C I^{1-1/p}\|\,|x|^{\al+1}Du\|\_p. \tag{20}\label{20}
\end{equation\*}
Using now conditions $u\in C\_0^1(\R^n)$ and \eqref{10} again, we see that
\begin{equation\*}
 I\le C\_2\int\_{\R^n}dx\,|x|^{p\al}1(|x|\le C\_2)=C\_3\int\_0^{C\_2} dr\, r^{p\al+n-1}<\infty,
\end{equation\*}
where $C\_2,C\_3$ are positive real numbers, possibly depending on $u$.
Thus, by \eqref{5} and \eqref{20},
\begin{equation\*}
 \|\,|x|^\al u\|\_p\le C\| |x|^{\al+1}Du\|\_p. \quad\Box
\end{equation\*}


