Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic $L$ log $L$ and $A_\infty$ constants

Abstract

In the dyadic case the union of the Reverse Hölder classes, p>1 RH d p is strictly larger than the union of the Muckenhoupt classes p>1 A d p = A d ∞ . We introduce the RH d 1 condition as a limiting case of the RH d p inequalities as p tends to 1 and show the sharp bound on RH d 1 constant of the weight w in terms of its A d ∞ constant. We also take a look at the summation conditions of the Buckley type for the dyadic Reverse Hölder and Muckenhoupt weights and deduce them from an intrinsic lemma which gives a summation representation of the bumped average of a weight. Our lemmata also allow us to obtain summation conditions for continuous Reverse Hölder and Muckenhoupt classes of weights and both continuous and dyadic weak Reverse Hölder classes. In particular, it shows that a weight belongs to the class RH 1 if and only if it satisfies Buckley's inequality. We also show that the constant in each summation inequality of Buckley's type is comparable to the corresponding Muckenhoupt or Reverse Hölder constant.To prove our main results we use the Bellman function technique. I. Definitions and Main Results.Recently different approaches to dyadic and continuous A ∞ class gave an essential improvement of the famous A 2 conjecture. The improvement, called A p − A ∞ bound for Calderon-Zygmund operators, was obtained by means of the observation that if a weight w belongs to the Muckenhoupt class A p , then it belong to a bigger class A ∞ , and a certain sequence satisfies the Carleson property. We refer the reader to papers [HPTV], [HyPer] for the precise proof of A 2 − A ∞ bound (in [HPTV] it is not formulated, but can be seen from the proof), and to [HyLa] for a full proof of the A p − A ∞ bound.Carleson sequences, related to A p weights, appeared in many papers, where boundedness of singular operators was studied. Many of them were proved using Bellman function method. Using this method, the Carleson embedding theorem was proved in [NTV1]. Results related to Carleson measures (partially proved with certain Bellman functions) also appeared in [NTV2], [Wit], [PP]. Also, the "easy" case of the two weight inequality, [VaVo2], is a certain summation condition, and was also obtained by means of Bellman function. Most of our proofs will use very natural (but not totally sharp) Bellman functions.Let us explain our results in more details. In this paper we present equivalent definitions of Muckenhoupt classes A p , Reverse Hölder classes RH p , and prove sharp inequalities, that show that these definitions are indeed equivalent. One type of these definitions is given in terms of Carleson sequences. Also, we define limiting cases A ∞ and RH 1 , which in the continuous case appear to be same sets (see [BR]), but in the dyadic case the class RH 1 is strictly bigger. We give equivalent definitions of these classes in terms of certain Carleson sequences; besides this, we give a sharp estimate on so called A ∞ and RH 1 constants, which appears to be much harder than the continuous case (and, actually, somehow uses the continuous res...

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