On discrete weighted Hardy type inequalities and properties of weighted discrete Muckenhoupt classes

In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponents as well as the new constants of the new classes. As an application, we establish the self-improving property (forward propagation property) of the discrete Gehring class.


Introduction
The study of regularity and boundedness of the discrete operators on p and higher summability of sequences was considered in the literature; see for example [2, 19-21, 28, 29] and the references cited therein. We also refer the reader to the papers [5-8, 28, 29, 33-35] for more results in the discrete spaces. In this paper, we confine ourselves to proving the self-improving property of the weighted discrete Muckenhoupt classes and then derive the self-improving property of the discrete Gehring classes. For the structure and relations between classical Muckenhoupt and Gehring classes (in integral forms) and their applications, we refer the reader to [1, 3, 9-11, 13-18, 22-26], [27,30], and [36].
Throughout the paper, we assume that 1 < p < ∞ and I ⊆ Z + is a fixed interval and the cardinality of a set J ⊂ I will be denoted by |J|. A discrete weight v defined on Z + = {1, 2, . . .} is a sequence v = {v(n)} ∞ n=1 of nonnegative real numbers. A sequence of positive real numbers λ will be considered the weight in the space p λ (Z + ) and the norm will be considered as A discrete nonnegative weight v belongs to the discrete Muckenhoupt class A p (A) on the interval I ⊆ Z + for p > 1 and A > 1 if the inequality holds for every subinterval interval J ⊂ I. For a given exponent p > 1, we define the A pnorm of the discrete weight v by the following quantity: where the supremum is taken over all intervals J ⊂ I. By fixing a constant A > 1, the pair of real numbers (p, A) defines a discrete Muckenhoupt class A p (A) as follows: and we will refer to A as the A p -constant of the class. Note that from Hölder's inequality [v] A p ≥ 1 holds for all 1 < p < ∞ and the following inclusion is true: In [32], the authors proved that v ∈ A p , 1 < p < ∞ if and only if v -1 p-1 ∈ A p , where p is the conjugate of p, and if v ∈ A p (1 < p < ∞), then v ∈ A q for every q > p and that v α ∈ A p for any 0 < α < 1. Böttcher and Seybold [4] proved that if v ∈ A p then there is ε = ε p,v such that v r ∈ A p for all r ∈ (1ε, 1 + ε). Böttcher and Seybold also proved the self-improving property for the discrete class A p of weights which states that: if v ∈ A p (C), then there exist constants δ > 0 and C 1 > 0 such that v ∈ A p-δ (C 1 ), or equivalently holds, without obtaining the exact values of δ and C 1 . For a given exponent q > 1 and a constant K > 1, a discrete nonnegative weight v belongs to the discrete Gehring class G q (K) (or satisfies a reverse Hölder inequality) on the interval For a given exponent q > 1, we define the G q -norm of v as where the supremum is taken over all intervals J ⊂ I and represents the best constant for which the G q -condition holds true independently on the interval J ⊆ I. We say that v is a discrete Gehring class G q -weight if its G q -norm is finite, i.e., Note that by Hölder's inequality, [v] G q ≥ 1 holds for all 1 < q < ∞ and the following inclusion is true: In [4], Böttcher and Seybold proved that if v ∈ A p (C), then there exist constants δ > 0 and K 1 < ∞ depending only on p and v such that for all ε ∈ [0, δ] and all J of the form |J| = 2 r with r ∈ N (the set of natural numbers). Note that inequality (1.7) is the reverse of the Hölder inequality so the result of Böttcher and Seybold is: if v ∈ A p (C), then v ∈ G q (C 1 ), and then holds for every subinterval J ⊂ I and |J| is the cardinality of the set J. In [29], the authors proved that if v is a nonnegative, nonincreasing sequence satisfying (1.9) for A > 1, then the inequality then v ∈ G p (A 1 ), and so That is, every Muckenhoupt A 1 -weight belongs to some Gehring class (a transition property). Later, in [31], the authors improved these results by giving sharp explicit values of the exponents p and A 1 .
Now, we give the definition of the weighted Muckenhoupt class A p λ (C), with a weight λ, that will be considered in this paper. A nonnegative discrete weight v defined on a a fixed interval I is called an ≥ 1 for all 1 < p < ∞ and the following inclusion is true: Our aim in this paper is to first prove some new refinements of discrete weighted inequalities with negative powers on finite intervals, then we use these inequalities to prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds. In particular, we prove that if v ∈ A p λ (C), then there exist constants δ, As an application, we deduce the self-improving property of the discrete Gehring class, i.e., we prove that if v ∈ G p (K), then there exist constants , K 1 > 0 such that v ∈ G p+ (K 1 ). The technique used in this paper allows us to get exact values of the exponents (δ and ) and also exact values of the constants C 1 and K 1 . These values correspond to the sharp values obtained by Nikolidakis (see [26] and [25]) in the continuous case.
The paper is organized as follows: In Sect. 2, we prove some fundamental inequalities of Hardy's type with negative powers that will be needed in the proof of the main results. In Sect. 3, we prove the main results and show how to obtain the self-improving property of the discrete Gehring class by using the discrete weighted Muckenhoupt class.

Main inequalities
In this section, we prove some fundamental inequalities that shall be used in the proof of the main results. Throughout the paper, we assume that the weights in study are nonnegative sequences defined and use the conventions 0 · ∞ = 0 and 0/0 = 0 and b k=m y(k) = 0, whenever m > b. We fix an interval J ⊂ Z + and consider subintervals J of the form {1, 2, . . . , N} for 1 < N ≤ ∞ and assume that λ is a positive sequence defined on J. The classical Hölder inequality asserts that where u and v are real sequences defined on J, p > 1 and 1 p + 1 q = 1. The summation by parts formula reads where is the forward difference operator defined by u(n) = u(n + 1)u(n). For any sequence v : J → R + (a nonnegative sequence), we define the operator (weighted Hardy Also, we have by using the above inequality that From these two facts, we have the following properties of H λ v.
Remark 2.1 As a consequence of Lemma 2.1, we notice that if v is nonnegative and nondecreasing, then H λ v q ≤ v q . We also notice from Lemma 2.1 that if v is nonnegative and nondecreasing, then H λ v q is also nonnegative and nondecreasing for q > 1.
Also, from the definition of H λ , we see that if v is nonincreasing, then Also, we have by using the above inequality that From these two facts, we have the following properties of H λ v.
Remark 2.2 As a consequence of Lemma 2.2, we notice that if v is nonnegative and nonincreasing, then H λ v q ≥ v q . We also notice from Lemma 2.2 that if v is nonnegative and nonincreasing, then H λ v q is also nonnegative and nonincreasing for q > 1.
Proof First we consider the case when p = q and prove that the inequality -p holds for p > 0. For brevity, we write (k) = H λ v(k). Now, by employing summation by parts with u(k) = λ(k), v(k + 1) = ( (k)) -p , we have and then we obtain By applying the inequality with γ = -p < 0, y = (k) and z = (k -1), and taking into account that the nondecreasing property of v implies that (k) = H λ v(k) is also nondecreasing and then y ≥ z, then we have This implies that which proves inequality (2.4) when p = q. Now, consider the case when p = q and fix q ∈ (0, p). Then, by applying Hölder's inequality with exponents p/q > 1 and p/(pq) and a weight λ and then by using (2.10), we get that Using the substitution (k) = H λ v(k), we obtain the desired inequality (2.4). The proof is complete.
By employing the last inequality with y = y 1 /y 2 , where y 1 , y 2 > 0, we get that (2.13) Let By using these values in (2.13), we get that Thus, we have By multiplying both sides by λ(k) and summating from 1 to n, we have By applying Theorem 2.1 on the last term and simplifying, (2.14) becomes which is the desired inequality (2.11). The proof is complete.
As a special case, when p = q = r, r > 0, we obtain the following result.

Theorem 2.4
Assume that v is a nondecreasing weight. If p ≥ q > 0, then Proof We proceed as in the proof of Theorem 2.1 to obtain (2.9), which writes By simplifying, we conclude that Fix q ∈ (0, p), by applying Hölder's inequality with exponents p/q > 1 and p/(pq) and a weight λ, and then by using inequality (2.17), we obtain . (2.18) Now, in order to complete the proof, we use the inequality which is a variant of the well-known Bernoulli inequality. This inequality is valid for all u ≥ 0 and u + v ≥ 0 or u > 0 and u + v > 0, and equality holds if and only if v = 1. Now, by employing (2.19) with γ = q/p < 1, and noting that (see (2.17)) By substituting the last inequality into (2.18), we obtain from which we obtain the desired inequality (2.16). The proof is complete.

Theorem 2.5
Assume that v is a nondecreasing weight. If p ≥ q > 0, then Proof We proceed as in the proof of Theorem 2.2 to obtain By applying Theorem 2.4, we get and then (2.21) becomes By combining similar terms, we obtain which is the desired inequality (2.20). The proof is complete.
The following lemma plays a crucial rule in establishing our main results.

Lemma 2.3
Let p > 1, v be nonincreasing weight on I, and let H λ v be defined by (2.3). If γ > 1, then holds for all n ∈ I.
Proof By applying Lemma 2.4 with γ = (p -1)/(q -1) > 1 for p > q > 1, we obtain Since v ∈ A p λ (C), we see that Consider the function That is, φ is decreasing for all β > α. Now, choose Since v is nondecreasing, we see that v -1 p-1 is nonincreasing, and then from Lemma 2.2 we have that p-1 , which implies that α ≤ β. Also, by using (3.3), we see that β ≤ ζ . Thus, we have α ≤ β ≤ ζ , and then This implies that From the last inequality and (3.4), we get that This leads to (3.5) By replacing p and q with 1/(q -1) > 0 and 1/(p -1) > 0, respectively, in inequality (2.11), we obtain By combining (3.5) and (3.6) and simplifying, we have Clearly, ψ(p) = 1 > 0 and since -φ(p, x) is a strictly increasing function for positive values of x, then the same holds for ψ(x) which will vanish for a certain value q 0 < p given by the unique positive solution of the equation φ(p, x) = 1. Then ψ(q 0 ) = 1, and ψ(x) > 0 ⇔ φ(p, x) < 1.
Since v is nondecreasing, then we get (from Lemma 2.1) that and since q -1 < p -1, then (3.8) gives By using this in (3.7), we have 1 -pq p -1 (Cq) which implies that That is v ∈ A q λ (C 1 ). The proof is complete.
Now, we will refine the above result by improving the constant that appears in (3.2).
Theorem 3.2 Assume that p > 1 and v is nondecreasing on the interval J. If v ∈ A p λ (C), then v ∈ A q λ (C 2 ) for any q ∈ (q 0 , p], where q 0 is the unique root of equation (3.1). Furthermore the constant C 2 is given by where p,q (C) := 1 -(Cq) 1 p-1 ( p-q p-1 ) > 0.