Vol. 9, No. 2, 2016

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Mutual estimates for the dyadic reverse Hölder and Muckenhoupt constants for the dyadically doubling weights

Oleksandra V. Beznosova and Temitope Ode

Vol. 9 (2016), No. 2, 307–316
Abstract

Muckenhoupt and reverse Hölder classes of weights play an important role in harmonic analysis, PDEs and quasiconformal mappings. In 1974, Coifman and Fefferman showed that a weight belongs to a Muckenhoupt class Ap for some 1 < p < if and only if it belongs to a reverse Hölder class RHq for some 1 < q < . In 2009, Vasyunin found the exact dependence between p, q and the corresponding characteristic of the weight using the Bellman function method. The result of Coifman and Fefferman works for the dyadic classes of weights under an additional assumption that the weights are dyadically doubling. We extend Vasyunin’s result to the dyadic reverse Hölder and Muckenhoupt classes and obtain the dependence between p, q, the doubling constant and the corresponding characteristic of the weight. More precisely, given a dyadically doubling weight in RHpd on a given dyadic interval I, we find an upper estimate on the average of the function wq (with q < 0) over the interval I. From the bound on this average, we can conclude, for example, that w belongs to the corresponding Aq1d-class or that wp is in Aq2d for some values of qi. We obtain our results using the method of Bellman functions. The main novelty of this paper is how we use dyadic doubling in the Bellman function proof.

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Keywords
reverse Hölder, Muckenhoupt, weights, dyadic
Mathematical Subject Classification 2010
Primary: 42B37
Milestones
Received: 5 November 2014
Revised: 30 March 2015
Accepted: 16 April 2015
Published: 2 March 2016

Communicated by Kenneth S. Berenhaut
Authors
Oleksandra V. Beznosova
Department of Mathematics
Box 870350
University of Alabama
Tuscaloosa, AL 35487-0350
United States
Temitope Ode
Baylor University
Waco, TX 76798
United States