Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant

Let $$\tilde\kappa$$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $$\tilde Q$$ and target distribution $$\tilde\mu$$ (see definitions below).

I want to choose $$(w_i)_{i\in I}$$ such that there is a $$\beta\in[0,1)$$ as small as possible with $$\left\|\tilde\kappa(\tilde g-\tilde\mu\tilde g)\right\|_{L^2(\tilde\mu)}\le\beta\left\|\tilde g-\tilde\mu\tilde g\right\|_{L^2(\tilde\mu)}\tag1$$ for all nonnegative $$g\in\mathcal L^2(\mu)$$ and $$\tilde g(i,x'):=(g\circ\varphi_i)(x')\;\;\;\text{for }(i,x')\in\tilde E.$$

Fix any such $$\tilde g$$. Note that $$\int\tilde g\:{\rm d}\tilde\mu=\int g\:{\rm d}\mu$$ and $$(\tilde\kappa\tilde g)(\tilde x)-\int\tilde g\:{\rm d}\tilde\mu=\int\tilde\lambda({\rm d}\tilde y)\left(\tilde k(\tilde x,\tilde y)-\tilde p(\tilde y)\right)\left(\tilde g(\tilde y)-\tilde g(\tilde x)\right)\tag2.$$

I guess we first need to show that there exists (at least one) choice of $$(w_i)_{i\in I}$$ such that there is a $$\beta$$ satisfying $$(1)$$ and then try to minimize $$\beta$$ over all possible choices of $$(w_i)_{i\in I}$$. How can we possibly do that?

Substituting the definitions, we may note that $$$$\begin{split}&\left\|\tilde\kappa(\tilde g-\tilde\mu\tilde g)\right\|_{L^2(\tilde\mu)}^2\\&\;\;\;\;=\sum_{i\in I}\int_{\{\:w_ip\:>\:0\:\}}\mu({\rm d}x)w_i(x)\left|\sum_{j\in I}\int\lambda({\rm d}y)\left(q_j\wedge\frac{w_j(y)p(y)}{w_i(x)p(x)}q_i(x)\right)\sigma_{ij}(x,y)(g(y)-g(x))+g(x)-\mu g\right|^2.\end{split}\tag3$$$$

Definitions: Let

• $$(E,\mathcal E,\lambda)$$, $$(E',\mathcal E',\lambda')$$ be measurable spaces;
• $$I$$ be a finite nonempty set;
• $$p,q_i$$ be probability densities on $$(E,\mathcal E,\lambda)$$ for $$i\in I$$;
• $$w_i:E\to[0,1]$$ be $$\mathcal E$$-measurable with $$\sum_{i\in I}w_i=1$$;
• $$\varphi_i:E'\to E$$ be bijective and $$(\mathcal E',\mathcal E)$$-measurable with $$\lambda'\circ\varphi_i^{-1}=q_i\lambda$$ for $$i\in I$$;
• $$w'_i:=w_i\circ\varphi_i$$ and $$p'_i:=\frac p{q_i}\circ\varphi_i$$ for $$i\in I$$;
• $$\zeta$$ denote the counting measure on $$(I,2^I)$$;
• $$(\tilde E,\tilde{\mathcal E},\tilde\lambda):=(I\times E',2^I\otimes\mathcal E',\zeta\otimes\lambda')$$
• $$\tilde p:=w'p'$$ and $$\tilde\mu:=\tilde p\tilde\lambda$$;
• $$\tilde q:\tilde E^2\to[0,\infty)$$ be symmetric and $$\tilde{\mathcal E}^{\otimes2}$$-measurable with $$\int\tilde\lambda({\rm d}\tilde y)\tilde q(\tilde x,\tilde y)=1$$ for all $$\tilde x\in\tilde E$$ and $$\tilde Q(\tilde x,\;\cdot\;):=\tilde q(\tilde x,\;\cdot\;)\tilde\lambda$$;
• $$\tilde\alpha(\tilde x,\tilde y):=1\wedge\frac{\tilde p(\tilde y)}{\tilde p(\tilde x)}$$ for $$\tilde x,\tilde y\in\tilde E$$;
• $$\tilde k:=\tilde\alpha\tilde q$$;
• $$\sigma_{ij}(x,y):=\tilde q((i,\varphi_i^{-1}(x)),(j,\varphi_j^{-1}(y)))$$ for $$(i,x),(j,y)\in I\times E$$;
• $$\tilde\kappa(\tilde x,\tilde B):=\int_{\tilde B}\tilde Q(\tilde x,{\rm d}\tilde y)\tilde\alpha(\tilde x,\tilde y)+\left(1-\int\tilde Q(\tilde x,{\rm d}\tilde y)\tilde\alpha(\tilde x,\tilde y)\right)1_{\tilde B}(\tilde x)$$ for $$(\tilde x,\tilde B)\in\tilde E\times\tilde{\mathcal E}$$.