Asymptotically sharp A_∞ to A_p embedding for flat weights with [w]_{A_∞} near 1, yielding quantitative weighted Poincaré-Sobolev inequalities that approach the unweighted case.
Kinnunen, Sharp results on reverse H \"older inequalities , Ann
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Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincar\'e-Sobolev inequalities
Asymptotically sharp A_∞ to A_p embedding for flat weights with [w]_{A_∞} near 1, yielding quantitative weighted Poincaré-Sobolev inequalities that approach the unweighted case.