in a ball (case $1\leqslant p < n$)
Let $f\in W^1_p(\mathbb R^n)$, $1\leqslant p < n$ and $p^* = \frac{np}{n-p}$ then the following inequality holds
\begin{equation}
\Bigl(\int\limits_{B}|f(x)-f_B|^{p^*}\,dx\Bigr)^{\frac{1}{p^*}} \leqslant C\Bigl(\int\limits_{B}|\nabla f(x)|^{p}\,dx\Bigr)^{\frac{1}{p}}
\end{equation}
for any balls $B \subset \mathbb R^n$, and constant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.
in a ball (case $1\leqslant p < \infty$)
There is a weaker inequality which is derived from previous one by inserting the measure of ball B and applying Hölder inequality.
\begin{equation}
\frac{1}{|B|}\int\limits_{B}|f(x)-f_B|^{p}\,dx \leqslant \frac{Cr^p}{|B|}\int\limits_{B}|\nabla f(x)|^{p}\,dx,
\end{equation}
where $r$ denotes the radius $B$.
\begin{equation}
\frac{1}{|B|}\int\limits_{B}|f(x)-f_B|^{p}\,dx \leqslant \frac{Cr^p}{|B|}\int\limits_{B}|\nabla f(x)|^{p}\,dx,
\end{equation}
where $r$ denotes the radius $B$.
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