Asymptotically Sharp Reverse Hölder Inequalities for Flat Muckenhoupt Weights

• Published in: Indiana University mathematics journal Vol. 67; no. 6; pp. 2363 - 2391
• Main Authors: Parissis, Ioannis, Rela, Ezequiel
• Format: Journal Article
• Language: English
• Published: Department of Mathematics of Indiana University 01-01-2018
• ISSN: 0022-2518; 1943-5258
• DOI: 10.1512/iumj.2018.67.7522

We present reverse Hölder inequalities for Muckenhoupt weights in ℝn with an asymptotically sharp behavior for flat weights, namely, A∞ weights with Fujii-Wilson constant (W)A∞ → 1⁺. That is, the local integrability exponent in the reverse Hölder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Hölder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Hölder inequalities for one-sided and twosided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Hölder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.