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.
The two-weight fractional Poincar\'e-Sobolev sandwich
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abstract
We establish a two-weight fractional Poincar\'e-Sobolev sandwich, consisting of a two-weight fractional Poincar\'e-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches $1$. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin difference norms. Our methods unify, simplify and significantly extend various earlier approaches.
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math.CA 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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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.