Weighted logarithmic Sobolev inequality

Question

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that $$ \Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu $$ where the entropy $$ \Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int f^2 d\mu}\right) d\mu. $$ There have been generalizations for weighted inequalities, i-e of the form $$ \Ent_\mu(f^2)\leq C\int \omega|\nabla f|^2 d\mu $$ for some weight $\omega(x)$, see for example this paper and references therein. However it seems that people are mostly concerned with growing weights, $1\leq \omega(x)\to+\infty$ as $|x|\to\infty$ in the whole space. For applied models that I am currently dealing with, I am interested in the opposite case of bounded domains $\Omega\subset \mathbb R^d$ with weights $\omega(x)>0$ decaying on the boundary $$ \omega(x)\to 0 \qquad\mbox{as }x\to\partial\Omega. $$ (Typically $\omega(x)\sim\operatorname{dist}(x,\partial\Omega)$). I find it extremely difficult to navigate the immense bibliography on this topics, can anyone point me to a first anchor point for the specific case of vanishing weights as above? Any pointer, direction, bibliographical entry point would be extremely useful.