Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincar\'e-Sobolev inequalities

We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1 when the weight is nearly constant. As intermediate steps towards the result, we obtain quantitative estimates on the weighted and unweighted BMO norms of $\log w$ for an $A_\infty$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincar\'e-Sobolev inequality can be proved for weights with small $[w]_{A_\infty}$ that recovers the classical Sobolev exponent $p^*=\frac{np}{n-p}$ when $[w]_{A_\infty}\to 1^+$.