It is a well-known fact that the union p>1 RH p of the Reverse Hölder classes coincides with the union p>1 A p = A ∞ of the Muckenhoupt classes, but the A ∞ constant of the weight w, which is a limit of its A p constants, is not a natural characterization for the weight in Reverse Hölder classes. In the paper, the RH 1 condition is introduced as a limiting case of the RH p inequalities as p tends to 1, and a sharp bound is found on the RH 1 constant of the weight w in terms of its A ∞ constant. Also, the sharp version of the Gehring theorem is proved for the case of p = 1, completing the answer to the famous question of Bojarski in dimension one.The results are illustrated by two straightforward applications to the Dirichlet problem for elliptic PDE's.Despite the fact that the Bellman technique, which is employed to prove the main theorems, is not new, the authors believe that their results are useful and prove them in full detail. §1. Definitions and main resultsWe say that w is a weight if it is a locally integrable function on the real line, positive almost everywhere (with respect to the Lebesgue measure). Let m J w be the average of a weight w over a given interval J ⊂ R:A weight w belongs to the Muckenhoupt class A p whenever its Muckenhoupt constant [w] A p is finite:Note that, by Hölder's inequality, [w] A p ≥ 1 for all 1 < p < ∞, and that the following inclusion is true:So, for 1 < p < ∞ the Muckenhoupt classes A p form an increasing chain. There are two natural limits of it -as p approaches 1 and as p goes to ∞. We shall be interested in the limiting case as p → ∞, A ∞ = p>1 A p . There are several equivalent definitions of it. We state the one that we are going to use (the natural limit of the A p conditions, which also defines the A ∞ constant of the weight w); for other equivalent definitions, see [GaRu, Gr] or [St93]:2010 Mathematics Subject Classification. Primary 42B20.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 28 O. BEZNOSOVA AND A. REZNIKOV m I w .We want to make a remark about this definition.Remark 1. Inequality (1.4) can be rewritten in the following way:Note that, since the function x log x is concave, by Jensen's inequality we also haveCondition (1.4) is actually much more natural whenever one deals with Reverse Hölder conditions rather than with A p conditions, see, for example, [Fe,Cor07,HyPer].There is no standard notation here, sometimes this class is called RH L log L , because (1.4) is the reverse Jensen inequality for the function x log x; otherwise, it is called G 1 to emphasize the contribution of Gehring to the study of the Reverse Hölder classes. Sometimes, for the RH 1 constant one takes sup J⊂R exp m J w m J w log w m J w to remove the logarithm on the right-hand side of (1.5). We keep our notation because it is shorter and it is clear that we are working with the Reverse Hölder condition.

Abstract. Let B be a collection of open sets in R n such that, for any x ∈ R n , there exists a set U ∈ B of arbitrarily small diameter containing x. B is said to be a density basis provided that, given a measurable set A ⊂ R n , for a.e. x ∈ R n we haveholds for any sequence of sets {R k } in B containing x whose diameters tend to 0. The geometric maximal operatorIt is shown that the halo function associated to any homothecy invariant density basis is a continuous function on (1, ∞). However, an example of a homothecy invariant density basis is provided such that the associated halo function is not continuous at 1.

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...

The dyadic paraproduct is bounded in weighted Lebesgue spaces L p (w) if and only if the weight w belongs to the Muckenhoupt class A d p . However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L 2 (w) case. In this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L 2 (w) depends linearly on the A d 2 characteristic of the weight w using Bellman function techniques and extrapolate this result to the L p (w) case.

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