Higher integrability and reverse Hölder inequalities in the limit cases

We study the higher integrability of weights satisfying a reverse Hölder inequality (⨏ I uβ) 1 β ≤ B(⨏ I uα) 1 α for some B > 1 and given α < β, in the limit caseswhen α ∈ {−∞, 0} and/or β ∈ {0, +∞}. The results apply to the Gehring and Muckenhoupt weights and their corresponding limit classes.


Motivations and main results
A weight is a non-negative locally integrable function. Let u be a weight defined on a bounded interval I 0 ⊂ ℝ. For real and nonzero exponents α and β with α < β, we say that u verifies a reverse Hölder inequality if there exists a constant B such that, for every subinterval I ⊂ I 0 , where the barred integral stands for the integral mean 1 |I| ∫ I u. In this case, we write u ∈ RH α,β (B). The notation recalls that the reverse of inequality (1.1) is always true for B = 1 thanks to the Hölder's inequality. Therefore, in (1.1), necessarily, B ≥ 1, and the equality prevails in the case of constant functions. The quantity which makes the quantity RH α,β (u) well defined for all α, β ∈R = ℝ ∪ {−∞, +∞} with α < β. These classes have been introduced by Bojarski in [1], and their interest is based on the fact that they include the Muckenhoupt classes and the Gehring classes as particular cases, being Indeed, it turns out that the class RH α,β seems to be the right environment to study the main properties of A p and G q weights (see [13,14,19,20,24,28]): • self-improving of integrability: there exist s p and t q such that A p (w) < ∞ ⇒ A s (w) < ∞ for all s such that s p < s ≤ p, (1.6) G q (v) < ∞ ⇒ G t (v) < ∞ for all t such that q ≤ t < t q ; (1.7) • transition: having ⋃ p>1 A p = ⋃ q>1 G q , there exist σ p and τ q such that In fact, (1.6), (1.7), (1.8) and (1.9) can be proved to be special cases of the following higher integrability property enjoyed by weights satisfying condition (1.1): We refer to the optimal bounds γ + and γ − as the sharp higher integrability exponents, and to RH ν,β (u) and RH α,τ (u) as the sharp higher integrability constants. The problem of finding these sharp bounds has been fully solved in dimension one (see [13,20,28]). In particular, in [20] (see Section 2, Theorem 2.1), it has been proved that all those bounds can be obtained by mean of the unique two solutions to the following equation: where ω is an auxiliary continuous function defined by which is strictly increasing for x < min{0, α} and strictly decreasing for x > max{0, β}.
In a paper of 2009 [14], Korenovskii and Fomichev studied the higher integrability property (1.10) when α, β assume the limit values 0 or ±∞. They proved, case by case, that the sharp higher integrability exponents for functions in the limit classes can be obtained by passing to the limit as appropriate in the equation (1.11).
The aim of this paper is to complete their work by obtaining also the sharp higher integrability constants for the same limit classes.
In order to present the result, let us first show how it reads in the special case of Muckenhoupt weights by recalling the relationships existing among the constants A 1 (w), A p (w) and A ∞ (w). Namely, the known theorem proved by Sbordone and Wik [25] states that and more recently, it has been proved that (at least) in dimension one, a corresponding property holds true for the class A 1 (see [21]), i.e., Therefore, it is natural to ask if similar connections hold true for the Gehring classes and in more general classes RH α,β . The answer is given by the following theorem. Theorem 1.1. Let α, β be real nonzero numbers such that α < β.
• If u belongs to the limit class RH L,β with L ∈ {−∞, 0}, then • If u belongs to the limit class RH α,L with L ∈ {0, +∞}, then It is easy to observe that A 1 (u) = RH −∞,1 (u) and A ∞ (u) = RH 0,1 (u) so that (1.12) and (1.13) are immediate corollaries of Theorem 1.1. Notice that this result is another step forward in the unification of the theory of Muckenhoupt and Gehring weights which provides a unique method and general theorems to study the entire set of classes (1.4) and (1.5).
The applications of the higher integrability properties of RHI weights cover several fields of analysis such as, to mention some examples, bounded maximal operator in weighted L p spaces [18], quasiconformal mappings [8], L p solvability of the Dirichlet problem [26], quasi-minimizers for one-dimensional Dirichlet integral [17] and many others. Good references on the theory are the books [9,13]. This paper will proceed as follows. In Section 2, we recall the general theorem on the sharp integrability in RHI classes and show how it is made possible to achieve a unified approach to Muckenhoupt classes. In Section 3, we study the monotonicity and continuity of sharp integrability exponents and the sharp integrability constants. In Sections 4 to 9, we study each limit class according to the structure of the paper of Korenovskii and Fomichev [14]. For the convenience of the reader, we adopt, as much as possible, their notations. Theorem 1.1 will be proved separately for each limit case, and we will highlight how it works for interesting special cases of Muckenhoupt and Gehring classes.

Optimal integrability from reverse Hölder inequalities and the unified theory of Muckenhoupt weights
To state the theorem on the optimal higher integrability of weights satisfying a reverse Hölder inequality, we need to extend the definition of the function ω in order to cover the limit values of α and β. We set With these positions, we can formulate the sharp higher integrability in RH α,β classes.
To show how this theorem provides a unification for the theory of Muckenhoupt weights, let us recall the definitions. The class A p is defined as the set of weights w such that In the limit cases p = 1 and p = ∞, the Muckenhoupt condition becomes, respectively, (for alternative definitions of A ∞ , see, for example, [10,23]). For these classes, the transition property (1.8) encodes the well-known fact that Muckenhoupt weights satisfy a reverse Hölder inequality and the Gehring lemma on higher integrability (a classical reference is [3]). There are many papers where sharp bounds and sharp constants for the embeddings of Muckenhoupt classes in Gehring classes are proved case by case (see [2, 4-6, 11, 12, 15, 16, 20, 27]). We point out that these results have been achieved by means of independent theorems, one for each of the classes above. Instead, by using Theorem 2.1, we can reformulate the sharp results for the transition property in a new unified shape, which shows the following interesting symmetries (see [20]).

Monotonicity and continuity of sharp self-improving bounds
We start this section by recalling the monotonicity of the constants RH α, The immediate consequence of the monotonicity is that, for α 1 ≤ α 2 and β 1 ≤ β 2 , we have In the limit cases the previous inequalities become Let us now define the self-improving sharp exponents as follows: We will refer to ν α,β and τ α,β , respectively, as the lower sharp exponent and the upper sharp exponent for the class RH α,β (B).
For β ∈ ℝ − {0}, a weight u belongs to the class RH −∞,β when, for every subinterval I ⊂ I 0 and for some B > 1 independent of the interval, it satisfies the inequality The ω-function for this class is In this case, the equation admits a unique positive solution γ + −∞,β ≥ β given by

Special case: the Muckenhoupt class A 1
In the special case α = −∞ and β = 1, with the change of variable α = 1 1−p , the class RH −∞,1 coincides with the well-known Muckenhoupt class A 1 which is the class of weights w such that, for A > 1 and for every subinterval I ⊂ I 0 , it is The constant RH −∞,1 (w) defined by (1.3) is usually denoted by The ω-function for this class becomes In this case, the unique solution is given by (see [2]) Clearly, we have α → −∞ ⇐⇒ p → 1 + , so in this case, Theorem 4.1 reads as follows.

Special case: the Gehring class G ∞
For α = 1 and β = +∞, the class RH 1,+∞ coincides with the class G ∞ which is the class of weights v such that, for G > 1 and for every subinterval I ⊂ I 0 , it is ess sup The ω-function for this class is which admits a unique positive solution given by .
In this class, Theorem 5.1 reads as follows.

Special case: the Muckenhoupt class A ∞
For α = 0 and β = 1, adopting again the change of variable α = 1 1−p , the class RH 0,1 coincides with the class A ∞ , the class of weights w such that, for every subinterval I ⊂ I 0 , for some constant A > 1, it is with the A ∞ -constant defined by The ω-function for this class is and the characteristic equation In this class, having again α → −∞ ⇔ p → +∞, Theorem 6.1 gives back the following well-known theorem.

Limit case 5: the class RH −∞,0
When α = −∞ and β = 0, we have the class of weights u such that, given a constant B > 1, for every subinterval I ⊂ I 0 , it is exp(⨏ I ln u) ≤ B ess inf I u.