The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates.

Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of Neumann and Dirichlet heat kernels on inner uniform domains **by showing the Poincaré inequality and the volume doubling property for the canonical Dirichlet forms on inner uniform domains.**

Namely, for inner uniform domains, we can check the Poincaré inequality and the volume doubling property.

It is easy to check that domains with cusps are not inner uniform domains. For example, $D=\{(x,y) \in \mathbb{R}^2 \mid x>0,\ y<1,\ y>x^{1/2}\}$ is not inner uniform.

The closure $\bar{D}$ of $D$ is regarded as a metric space endowed with the shortest path metric $\rho$.

I am concerned with whether the Poincaré inequality holds on $\bar{D}$. Namely, there exists $P_0$ such that for any $x \in \bar{D}$, $r>0$, and smooth $f$, \begin{align*} (\ast)\quad \inf_{\xi \in \mathbb{R}} \int_{B(x,r)}|f-\xi|^2\,dm \le P_o r^2 \int_{B(x,r)}|\nabla f|^2\,dm, \end{align*} where $B(x,r)=\{y \in \bar{D} \mid \rho(y,x)<r\}$ and $\nabla f$ denotes the distributional derivative of $f$. $m$ is the Lebesgue measure on $D$

**My question**

Does $(\ast)$ hold? The volume doubling property holds. Let $x$ be the origin and let $f$ be a smooth bump function with $\nabla f=1$, $f=1$ on $B(x,r/2)$, and $f=0$ outside $B(x,r)$. Then, RHS of $(\ast)=P_0 r^2 m(B(x,r)\setminus B(x,r/2)) \sim r^2 \times r\sqrt{r}=r^{7/2}$. What is the order of LHS of $(\ast)$?