$$\begin{aligned}
\label{e35}
 v_{\Omega,c_2}(x) \le
 v_{\Omega,V}(x)\le v_{\Omega,c_1}(x)\ ,\end{aligned}$$ and for any $0 <c < \infty$, $x\in \Omega$, $$\begin{aligned}
\label{e36}
 v_{\Omega,c}(x)\ge 2^{-2(3m+4)c/\lambda_1(\Omega)}\frac{\lambda_1(\Omega)}{8c+\lambda_1(\Omega)}\bigg(\frac{8c}{8c+\lambda_1(\Omega)}\bigg)^{8c/\lambda_1(\Omega)}v_{\Omega}(x).\end{aligned}$$ Hence $$\label{e37}
 \|v_{\Omega,c}\|_1\ge 2^{-2(3m+4)c/\lambda_1(\Omega)}\frac{\lambda_1(\Omega)}{8c+\lambda_1(\Omega)}\bigg(\frac{8c}{8c+\lambda_1(\Omega)}\bigg)^{8c/\lambda_1(\Omega)}\|v_{\Omega}\|_1,$$ and $$\label{e38}
 \|v_{\Omega,c}\|_{\infty}\ge 2^{-2(3m+4)c/\lambda_1(\Omega)}\frac{\lambda_1(\Omega)}{8c+\lambda_1(\Omega)}\bigg(\frac{8c}{8c+\lambda_1(\Omega)}\bigg)^{8c/\lambda_1(\Omega)}\|v_{\Omega}\|_{\infty}.$$ Furthermore the right-hand side of converges to $v_{\Omega}(x)$ as $c\downarrow 0$, and $$\label{e45a}
\int_{\Omega}dy\,p_{\Omega}(x,y;t)\le 2^{3m/4}e^{-t\lambda_1(\Omega)/4},\qquad \forall t>0,\, \forall x\in \Omega.$$

The two inequalities in follow immediately from , and the hypothesis $0\le c_1 \le V \le c_2$. To prove inequality , note that for any $T>0$, and any $c\ge 0$, $$\begin{aligned}
\label{e39}
v_{\Omega,c}(x)&=\int_{\Omega}dy\,\int_{{\mathbb{R}}^+}dt\, p_{\Omega,c}(x,y;t)\nonumber \\ &
=\int_{\Omega}dy\,\int_{{\mathbb{R}}^+}dt\, e^{-ct}p_{\Omega}(x,y;t)\nonumber \\ &
\ge \int_{\Omega}dy\,\int_0^Tdt\, e^{-ct}p_{\Omega}(x,y;t)\nonumber \\ &
\ge e^{-cT}\int_{\Omega}dy\,\int_0^Tdt\, p_{\Omega}(x,y;t)\nonumber \\ &
=e^{-cT}\Big(v_{\Omega}(x)-\int_T^{\infty}dt\,\int_{\Omega}dy\,p_{\Omega}(x,y;t)\Big).\end{aligned}$$ We need to estimate the double integral in the right-hand side of . By the heat semigroup property

$$p_{\Omega}(x,y;t)=\int_{\Omega}dz\,p_{\Omega}(x,z;t/2)p_{\Omega}(z,y;t/2)$$ we have $$\begin{aligned}
\label{e40}
\int_{\Omega}dy\, & p_{\Omega}(x,y;t) =\int_{\Omega}dy\,\int_{\Omega}dz\,p_{\Omega}(x,z;t/2)p_{\Omega}(z,y;t/2)\nonumber \\ &
= \int_{\Omega}dz\,p_{\Omega}(x,z;t/2)\int_{\Omega}dy\,\big(p_{\Omega}(z,y;t/2)\big)^{1/2}\big(p_{\Omega}(z,y;t/2)\big)^{1/2}.\end{aligned}$$ We estimate $p_{\Omega}(z,y;t/2)$ in terms of $\lambda_1(\Omega)$. Recall that $p_{\Omega}(x,y;t)$ can be expressed in terms of the eigenfunctions $\varphi_{j, \Omega}(x)$ by $$\label{e41}
p_{\Omega}(x,y;t) = \sum_{j =1}^{\infty} e^{- t\lambda_j(\Omega)} \varphi_{j, \Omega}(x)  \varphi_{j, \Omega}(y).$$ Moreover for any $\beta\in [0,1)$, $$\label{beta}
p_{\Omega}(x,x;t)\le e^{-\beta t\lambda_1(\Omega)}p_{\Omega}(x,x;(1-\beta)t).$$ The heat kernel for ${\mathbb{R}}^m$ is given by $$\label{e42}
p_{{\mathbb{R}}^m}(x,y;t)=(4\pi t)^{-m/2}e^{-|x-y|^2/(4t)},$$ and satisfies $$\label{e43}
\int_{{\mathbb{R}}^m}dy\,p_{{\mathbb{R}}^m}(x,y;t)=1,\qquad  \forall x\in {\mathbb{R}}^m,\, \forall t>0.$$ By the heat semigroup property, the Cauchy-Schwarz inequality, for 