Asymptotically sharp embedding of $A_{\mathrm{\infty }}$ into $A_p$ for flat weights and applications to Poincaré-Sobolev inequalities

We provide new quantitative results on the embedding of the Muckenhoupt class $A_{\mathrm{\infty }}$ into $A_p$ with the correct asymptotic behavior when the Fujii–Wilson constant $[w{]}_{A_{\mathrm{\infty }}}$ 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 $A_{\mathrm{\infty }}$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincaré-Sobolev inequality can be proved for weights with small $[w{]}_{A_{\mathrm{\infty }}}$ that recovers the classical Sobolev exponent $p^{\ast }=\frac{np}{n-p}$ when $[w{]}_{A_{\mathrm{\infty }}}\to 1^{+}$.