On structure of discrete Muchenhoupt and discrete Gehring classes

In this paper, we study the structure of the discrete Muckenhoupt classAp(C) and the discrete Gehring class Gq(K). In particular, we prove that the self-improving property of the Muckenhoupt class holds, i.e., we prove that if u ∈Ap(C) then there exists q < p such that u ∈ A(C1). Next, we prove that the power rule also holds, i.e., we prove that if u ∈Ap then uq ∈Ap for some q > 1. The relation between the Muckenhoupt class A1(C) and the Gehring class is also discussed. For illustrations, we give exact values of the norms of Muckenhoupt and Gehring classes for power-low sequences. The results are proved by some algebraic inequalities and some new inequalities designed and proved for this purpose.


Introduction
We fix an interval I ⊂ R and consider subintervals I of I and denote by |I| the Lebesgue measure of I. A weight w is nonnegative locally integrable function. In the literature a nonnegative measurable weight function w defined on a bounded fixed interval I is called an A p (C)-Muckenhoupt weight for 1 < p < ∞ if there exists a constant C < ∞ such that for every subinterval I ⊂ I. For a given exponent p > 1, we define the A p -norm of the function w by the following quantity: where the supremum is taken over all intervals I ⊂ I. For a given fixed constant C > 1, if the weight w belongs to A p (C), then A p (w) ≤ C. A weight w satisfying the condition is called an A 1 (C)-Muckenhoupt weight where C > 1. In [25] Muckenhoupt proved the following result. (3), then there exists p ∈ [1, C/(C -1)] such that

Lemma 1.1 If w is a nonincreasing weight satisfying condition
Bojarski, Sbordone, and Wik [3] improved the Muckenhoupt result by excluding the monotonicity condition on the weight w by using the rearrangement ω * of the function ω over the interval I and established the best constant. In particular, they proved the following lemma. (3) with C > 1, then there exists p ∈ [1, C/(C -1)] such that

Lemma 1.2 If w is a nonincreasing weight and satisfies condition
In [25] Muckenhoupt also proved the following result.

Lemma 1.3 If 1 < p < ∞ and w satisfies the A p -condition
(1) on the interval I, with constant C, then there exist constants q and C 1 depending on p and C such that 1 < q < p and w satisfies the A q -condition for every subinterval I ⊂ I.
In other words, Muckenhoupt's result (see also Coifman and Fefferman [9]) for selfimproving property states that: if w ∈ A p (C) then there exist a constant > 0 and a positive constant C 1 such that w ∈ A p-(C 1 ), and then Muckenhoupt [25] also proved the following result. Lemma 1.4 If 1 < p < ∞ and w ∈ A p (C) on the interval I with a constant C, then there exist constants r and C 1 depending only on p and C such that 1 < r and w r ∈ A p (C 1 ).
In recent years the study of the discrete analogues in harmonic analysis has become an active field of research. For example, the study of regularity and boundedness of discrete operators on l p analogues for L p -regularity and higher summability of sequences has been considered by some authors, see for example [2, 15-17, 26, 27] and the references they have cited. Whereas some results from Euclidean harmonic analysis admit an obvious variant in the discrete setting, others do not. The main challenge in such studies is that there are no general methods to study these questions, see for example [5-8, 26, 27, 30-32] and the references cited therein. We confine ourselves to proving the discrete analogue of Muckenhoupt results (Lemmas 1.1 and 1.3) and establish some inclusion properties between the discrete Muckenhoupt class and the discrete Gehring class. For structure and relations between classical Muckenhoupt and Gehring classes (in the integral forms) and their applications, we refer the reader to the papers [1, 3, 10-14, 18, 20-25, 28] and the references cited therein.
Throughout the paper, we assume that 1 < p < ∞ and I is a fixed finite interval from Z + . A discrete weight u defined on Z + = {1, 2, . . .} is a sequence u = {u(n)} ∞ n=1 of nonnegative real numbers. We consider the norm on l p (Z + ) of the form A discrete nonnegative weight u belongs to the discrete Muckenhoupt class A 1 (A) on the fixed interval I ⊂ Z + for p > 1 and A > 1 if the inequality holds for every subinterval J ⊂ I and |J| is the cardinality of the set J. A discrete nonnegative weight u 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 J ⊂ I. For a given exponent p > 1, we define the A p -norm of the discrete weight u by the following quantity: where the supremum is taken over all intervals J ⊂ I. Note that by Hölder's inequality A p (u) ≥ 1 for all 1 < p < ∞, and the following inclusion is true: For a given exponent q > 1 and a constant K > 1, a discrete nonnegative weight u belongs to the discrete Gehring class G q (K) (or satisfies a reverse Hölder inequality) on the interval I if, for every subinterval J ⊆ I, we have For a given exponent q > 1, we define the G q -norm of u as follows: 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. Note that by Hölder's inequality G q (u) ≥ 1 for all 1 < q < ∞, and that the following inclusion is true: Our aim in this paper, in the next section, is to prove the discrete analogy of the Muckenhoupt results which include the self-improving property of the Muckenhoupt class and we also prove the transition property due to Bojarski, Sbordone, and Wik [3] with a sharp constant. In particular, we prove that if u ∈ A p (C) then there exists q < p such that u ∈ A q (C 1 ) and if u ∈ A p then u q ∈ A p for some q > 1. For the relation between the discrete Muckenhoupt class and the discrete Gehring class, we prove that if u ∈ A 1 (C) then u ∈ G q (K) with exact values of the exponent q and the constant K. In addition, for illustration, we establish the exact values of the Muckenhoupt norm A q (n α ) and the Gehring norm G p (n α ) for power-low sequences {n α }.

Main results
Throughout this section, we assume that the sequences in the statements of theorems are nonnegative and assume for the sake of conventions that 0 · ∞ = 0, 0/0 = 0, b s=a y(s) = 0, whenever a > b, and We fix an interval I ⊂ Z + and consider I of the form {1, 2, . . . , k, . . . , N} (or [1, N] ⊂ Z + ). For any weight u : I → R + which is nonnegative, we define the operator Hu : I → R + by The following lemma gives some properties of the operator Hu that will be needed later. (17). Then we have the following properties: (1). If u is nonincreasing, then so is Hu(k) and Hu(k) ≥ u(k).

Lemma 2.1 Let Hu be defined as in
(2). If u is nondecreasing, then so is Hu(k) and Hu(k) ≤ u(k).
Proof (1). From the definition of H, we see that: If u is nonincreasing, then Hence, we have by using the above inequality that thus Hu(k) is nonincreasing. This completes the proof of the first case.
(2). If u is nondecreasing, then Also, we have by using the above inequality that thus Hu(k) is nondecreasing. This completes the proof of the second case. The proof is complete.
Remark 2.1 As a consequence of Lemma 2.1, we notice that if q > 1 and u is nonnegative, nonincreasing, then Hu q is also nonnegative and nonincreasing and Hu q ≥ u q . We also notice from Lemma 2.1 that if q > 1 and u is nonnegative and nondecreasing, then Hu q is also nonnegative and nondecreasing and Hu q ≤ u q .
In the proof of the next lemma, we shall use the notion of the characteristic function χ J defined on a set J by Lemma 2.2 Let 1 < q < ∞, and let u ∈ A q (C) for C > 1. Then, for any subset J = {1, 2, . . . , k}, we have that Proof For any nonnegative sequence λ(k) defined on I, we see for any subset J By using q = q/(q -1), we get that Multiplying both sides by (1/(k) q ) k s=1 u(s), we get that Now, since u ∈ A q (C), for C > 1, we see that By using (21) in (20), we get that For λ(s) = χ J (s), we see that k s=1 u(s) ≤ Ck sup s∈J u(s). That is, which is the desired inequality (19). The proof is complete.
Remark 2.2 The above lemma can be written as: if u ∈ A q (C) for some C > 1, then Hu(k) ≤ C sup k∈J u(k).

Lemma 2.3
Let 1 < q < ∞ and u be a nonincreasing weight. If u ∈ A q (C), then u ∈ A 1 (C).
Proof To prove the lemma, we need to prove that: if for some C > 1 independent of k, then for all 1 < s ≤ k, and k ∈ I. By using (22) and employing Lemma 3.1 in [29], we get that Now, by applying property (2) in Lemma 2.1 for the nondecreasing weight log u(s), we obtain that for all 1 < s ≤ k. The proof is complete.

Lemma 2.4 Let 1 < p < ∞ and u be a nonnegative weight. Then u ∈ A p if and only if
Proof From the definition of the class A p , and since 1p = 1/(1p) < 0, we have for A > 1 and all k ∈ I that The proof is complete.
The following lemma will play an important role in proving one of our main results.

Lemma 2.5 Assume that u is a nonincreasing weight, and let A(k) = k s=1 u(s). If p > 1, then
for all k ∈ I.
Proof Since u is nonincreasing, then so is ω(s) = A(s)/s, thus we have By applying Young's inequality ab ≤ a p p + b q q , a, b > 0 and 1 with q = p p-1 , a = ω(s) and b = ω p-1 (s), we obtain that By substituting (27) into (26), we obtain Using (28) and since 1-p p < 0, then we have Rewriting the last inequality and using ω = A(k)/k, we obtain that which is the required inequality (25). The proof is complete.
As a consequence of the above lemma and by the definition of H, and the fact that (Hu)(k) = A(k)/k, we obtain the following lemma.

Lemma 2.6
Assume that u is a nonincreasing weight, and let Hu be defined as in (17). If p > 1, then then, for r ∈ [1, C/(C -1)), we have that where A is given by Proof From the definition of Hu(k) and Lemma 2.5 with p = r > 1, we see that Define the function (η) = γ η r-1 -r -1 r η r for every γ > 0 and η ≥ γ .
By noting that, for η ≥ γ , we have That is, (η) is decreasing for η ≥ γ . From Lemma 2.1, we see that

Hu(s) ≥ u(s).
Now, by taking that γ = u(s), β = Hu(s) and θ = Cu(k), we see that γ ≤ β ≤ θ , and then we have This implies, by using (33), that By combining (32) and (34), we get that This implies that The proof is complete.
Remark 2.3 Theorem 2.1 is a discrete version of Lemma 1.2 and proves that if u ∈ A 1 (C) then u ∈ G r (A) for r ∈ [1, C/(C -1)) and a constant A given by (31).

Theorem 2.2
Let u be a nondecreasing weight. If 1 < p < ∞ and u ∈ A p (C), then there exist constants q and C 1 depending on p and C such that 1 < q < p and u ∈ A q (C 1 ).
Proof Since u ∈ A p (C), then it satisfies the condition From Lemma 2.4, we see also that u 1-p satisfies the A p -1 -condition Since 1p = -1/(p -1) and u is nondecreasing, we see that u -1 p-1 is nonincreasing. Now, applying Lemmas 2.2 and 2.1, we see that for r ∈ (1, r 0 ), with a constant A. Combining (35) and (36), we have that This shows that u satisfies the A q -condition, where q = 1 + (p -1)r and C 1 = A (p-1)/r C. It is immediate that q and C 1 depend only on C and p. The proof is complete.

Theorem 2.3
Let u be a nondecreasing weight on I with |I| = 2 r for r ∈ Z + . If 1 < p < ∞ and u satisfies the A p -condition (35) with constant C, then there exist constants q and C 1 depending on p and C such that 1 < q and u q satisfies the A p -condition with constant C 1 .
Proof In [4] Böttcher and Seybold proved that if u satisfies the A p -condition (35) with constant C, then there exists a constant m > 1 and C 1 < ∞ depending only on p such that for all m > 1 and all even natural numbers k. Now, by combining (36) and (37), we see that That is, No, let q = min{r, m}, then Hölder's inequality implies that and then (38) shows that where L = C 1/m 1 A (p-1)/r C. Taking the q th power, we get the desired result for p > 1. The proof is complete.
One of the basic special formulas in the differential calculus is the power rule (d/dt)t k = kt k-1 . Unfortunately, the difference of a power is complicated and not very useful since t n = (t + 1) nt n = n-1 k=0 n k t k .
In the following, we show how we can use the difference calculus to prove the property of the parameter of Muckenhoupt and Gehring classes for power-low sequences.

Conclusion
In this paper, we studied the structure of the discrete cases of the well-known Muckenhoupt class A p (C) and Gehring class G q (K). We established exact values of the norms of the discrete Muckenhoupt and Gehring classes for power-low sequences. The relations between the two classes have also been discussed. In fact we have proved that if the weight w belongs to the Muckenhoupt class A 1 (C), then it belongs to the same Gehring classes G q (K) for some q obtained from a solution of an algebraic equation.