Sharp estimates involving $A_\infty $ and $L\log L$ constants, and their applications to PDE

Abstract

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.

Theory of locally concave functions and its applications to sharp estimates of integral functionals

“…Estimates of integral functionals as provided by the Bellman function (1.2) lead to various sharp forms of the reverse Hölder inequality, see [34]. These cases were treated in the papers [1,4,24,33,34].…”

“…For convenience, we assume that I = [0, 1]. We note that the function t → ϕ 1 −1 [0,1] t 0 ϕ 1 (ζ) dζ is an increasing absolutely continuous function mapping [0, 1] to [0, 1]. Let ψ be its inverse function.…”

“…The paper [33] was the first to appear. Following this route, many authors have managed to prove various inequalities, see the papers [1,4,8,7,15,26,22,24,31,28,29,32,33,34,35,36,37,38]. Though the method turned out to be very useful, there was absolutely no theory that allowed to calculate the Bellman functions "mechanically".…”

Sharp estimates of integral functionals on classes of functions with small mean oscillation

“…As it has been said in the abstract, the particular cases of this problem were treated in the papers [1,2,4,5,6,9,10,11,12,14,15,16,18,19,20,21,22,23,24] (see Section 2 for a detailed explanation). The main reason for Problem 1.1 to be solvable (and it has been heavily used in all the preceeding work) is that the function B enjoys good properties.…”

Mixed Ap-A∞estimates with one supremum

“…The constant [w] A∞ was defined by Fujii [7] and Wilson [23,24], who also showed that both constants define the class A ∞ . Hytönen and Pérez [10] proved the quantitative upper bound (2) [w] A∞ [w] A exp ∞ and provided examples to show that [w] A exp ∞ can be exponentially larger than [w] A∞ (see also Beznosova and Reznikov [2]). While inequality (2) holds, it is not clear what the relationship is between A exp ∞ (w, Q) and A ∞ (w, Q) for a fixed cube Q. Hereafter we will refer to constants that contain a quantity depending on A exp ∞ (w, Q) as exponential A ∞ constants and constants depending on A ∞ (w, Q) as simply A ∞ constants.…”