$\beta=\frac12$, domain monotonicity of the Dirichlet heat kernel, and we have $$\begin{aligned}
\label{e44}
p_{\Omega}(z,y;t/2)&=\int_{\Omega}dw\,p_{\Omega}(z,w;t/4)p_{\Omega}(w,y;t/4)\nonumber \\ &
\le \bigg(\int_{\Omega}dw\,\big(p_{\Omega}(z,w;t/4)\big)^2\bigg)^{1/2}\bigg(\int_{\Omega}dw\,\big(p_{\Omega}(w,y;t/4)\big)^2\bigg)^{1/2}\nonumber \\ &
=\big(p_{\Omega}(z,z;t/2)\big)^{1/2}\big(p_{\Omega}(y,y;t/2)\big)^{1/2}\nonumber \\ &
\le e^{-t\lambda_1(\Omega)/4}\big(p_{\Omega}(z,z;t/4)\big)^{1/2}\big(p_{\Omega}(y,y;t/4)\big)^{1/2}\nonumber \\ &
\le e^{-t\lambda_1(\Omega)/4}\big(p_{{\mathbb{R}}^m}(z,z;t/4)\big)^{1/2}\big(p_{{\mathbb{R}}^m}(y,y;t/4)\big)^{1/2}\nonumber \\ &
=e^{-t\lambda_1(\Omega)/4}(\pi t)^{-m/2}.\end{aligned}$$ Substituting into we find $$\begin{aligned}
\label{e45}
\int_{\Omega}dy&\,p_{\Omega}(x,y;t)\le e^{-t\lambda_1(\Omega)/8}(\pi t)^{-m/4}\int_{\Omega}dz\,p_{\Omega}(x,z;t/2)\int_{\Omega}dy\,\big(p_{\Omega}(z,y;t/2)\big)^{1/2}\nonumber \\ &
\le e^{-t\lambda_1(\Omega)/8}(\pi t)^{-m/4}\int_{\Omega}dz\,p_{\Omega}(x,z;t/2)\int_{{\mathbb{R}}^m}dy\,\big(p_{{\mathbb{R}}^m}(z,y;t/2)\big)^{1/2}\nonumber \\ &
=2^{3m/4}e^{-t\lambda_1(\Omega)/8}\int_{\Omega}dz\,p_{\Omega}(x,z;t/2).\end{aligned}$$ Integrating both sides of with respect to $t$ over the set $[T,\infty)$ gives $$\begin{aligned}
\label{e46}
\int_T^{\infty}dt\,\int_{\Omega}dy\,p_{\Omega}(x,y;t)&\le 2^{3m/4}e^{-T\lambda_1(\Omega)/8}\int_T^{\infty}dt\,\int_{\Omega}dz\,p_{\Omega}(x,z;t/2)\nonumber \\ &
\le 2^{3m/4}e^{-T\lambda_1(\Omega)/8}\int_{{\mathbb{R}}^+}dt\,\int_{\Omega}dz\,p_{\Omega}(x,z;t/2)\nonumber \\ &
=2^{(4+3m)/4}e^{-T\lambda_1(\Omega)/8}v_{\Omega}(x).\end{aligned}$$ By combining and we find $$\label{e47}
v_{\Omega,c}(x)\ge e^{-cT}\big(1-2^{(4+3m)/4}e^{-T\lambda_1(\Omega)/8}\big)v_{\Omega}(x).$$ Choosing $T$ as to maximise the right-hand side of gives that $$\label{e48}
T=\frac{8}{\lambda_1(\Omega)}\log\Big(2^{(4+3m)/4}\Big(1+\frac{\lambda_1(\Omega)}{8c}\Big)\Big).$$ Inequality then follows by and . The inequalities and follow immediately from .

To prove we have by , $$\label{e45b}
p_{\Omega}(x,y;t)\le e^{-t\lambda_1(\Omega)/2}(2\pi t)^{-m/2}.$$ By the domain monotonicity, the non-negativity of the Dirichlet heat kernel, and , $$\begin{aligned}
\int_{\Omega}dy\,p_{\Omega}(x,y;t)&\le e^{-t\lambda_1(\Omega)/4}(2\pi t)^{-m/4}\int_{\Omega}dy\,\big(p_{\Omega}(x,y;t)\big)^{1/2}\nonumber \\ &
\le e^{-t\lambda_1(\Omega)/4}(2\pi t)^{-m/4}\int_{{\mathbb{R}}^m}dy\,\big(p_{{\mathbb{R}}^m}(x,y;t)\big)^{1/2}\nonumber \\ &
=2^{3m/4}e^{-t\lambda_1(\Omega)/4},\end{aligned}$$ which proves .

[*Proof of Theorem* ]{} Since $0\le V\le c$ we have by , $$\label{e56}
\| v_{\Omega,V} \|_1  \le \| v_{\Omega} \|_1,$$ and $$\label{e57}
\|v_{\Omega,V}\|_{\infty}\ge \|v_{\Omega,c}\|_{\infty}.$$ The upper bound follows from , , and .

Similarly, by , we have $$\label{e58}
\|v_{\Omega,V}\|_{\infty}\le \|v_{\Omega}\|_{\infty}.$$ and $$\begin{aligned}
\label{e59}
\|v_{\Omega,V}\|_1\ge\|v_{\Omega,c}\|_1.\end{aligned}$$ The lower bound follows from , , and . 