Poincaré inequality

Poincaré inequality in 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 $$\begin{equation}\label{eq:1} \Bigl(\int\limits_{B}|f(x)-f_B|^{p^*}\,dx\Bigr)^{\frac{1}{p^*}} \leqslant C\Bigl(\int\limits_{B}|\nabla f(x)|^{p}\,dx\Bigr)^{\frac{1}{p}} \end{equation}$$ for any balls $B \subset \mathbb R^n$, and constant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.

Poincaré inequality in ball (case $1\leqslant p < \infty$)

There is a weaker inequality which is derived from $\ref{eq:1}$ by inserting the measure of ball B and applying Hölder inequality.

$$\begin{equation}\label{eq:2} \frac{1}{|B|}\int\limits_{B}|f(x)-f_B|^{p}\,dx \leqslant \frac{Cr^p}{|B|}\int\limits_{B}|\nabla f(x)|^{p}\,dx, \end{equation}$$ where $r$ denotes the radius $B$.

References

[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[JH]	J. Heinonen, "Lectures on Analysis on Metric Spaces" Springer, New York, NY, 2001.