On Cesàro and Copson sequence spaces with weights

In this paper, we prove some properties of weighted Cesàro and Copson sequences spaces by establishing some factorization theorems. The results lead to two-sided norm discrete inequalities with best possible constants and also give conditions for the boundedness of the generalized discrete weighted Hardy and Copson operators.


Introduction
Recently the study of discrete spaces in functional and harmonic analysis has become an active field of research. For example, the study of discrete Cesàro space has been considered by some authors, see for example [6,22,27,29] and the references they cited. Whereas some results from Euclidean functional analysis admit an obvious variant in the discrete setting, some others do not. In fact, it is well known that passage from integral operators to their discrete analogues is not trivial, and each of these two settings requires its own techniques. In this paper, we study the structure of the weighted Cesàro and Copson sequence spaces. Throughout the paper, we assume that 1 ≤ p ≤ ∞. The Cesàro function space Ces p (R + ) is the set of all Lebesgue measurable real functions defined on R + = [0, ∞) such that The Cesàro function spaces Ces p (R + ) for 1 ≤ p ≤ ∞ were considered first by Shiue [26] and later by Hassard and Hussein [15], Sy, Zhang, and Lee [28], and Astashkin and Maligranda [4]. They proved that Ces p (R + ) for 1 < p < ∞ are separable Banach spaces and are not reflexive, and they do not have the fixed point property. They also proved that Ces ∞ (R + ) is a nonseparable Banach space. By the Hardy inequality [14] ∞ we see that the spaces L p (R + ) (i.e. space of all functions f such that ( ∞ 0 |f (x)| p dx) 1 p < ∞) are continuously embedded into Ces p (R + ) for 1 < p < ∞ with strict embedding. In other words, f ∈ Ces p (R + ) if and only if the operator T(f ) = (1/x) x 0 f (t) dt belongs to L p (R + ). Also, for 1 < p ≤ q < ∞, we see by the inequality (2) due to Opic and Kufner [24] that the spaces L p v (R + ) with weight v are embedded into Ces q (R + ) with weight u with strict embedding if It is clear that Hardy's inequality (1) can be interpreted as inclusions between the space of functions L p (R + ) and Cesàro space of functions Ces p (R + ); as a consequence, we get that The Cesàro sequence space ces p (N) is the set of all real sequences (λ n ) n≥1 on N that satisfy In 1968 the Dutch mathematical society posted a problem of finding an explicit representation of the discrete dual of the Cesàro space of sequences. In [26] Shiue investigated this problem for the first time, and later it was analyzed by Leibowitz [20] and Jagers [16], and for dual spaces of Cesàro space of sequences and functions, we refer to [29]. In particular, they proved that ces p (N) is a separable reflexive Banach space for 1 < p < ∞, and it does have the fixed point property, and if 1 < p < q < ∞, then ces p ⊂ ces q with continuous strict embedding. The representation result for the dual of the space of sequences of Cesàro type has been extended to the classical Lorentz space of sequences where λ * (n) is the nonincreasing rearrangement of |λ(n)| and q * is the conjugate of q by Arińo and Muckenhoupt [3]. They proved that the space d(v -q * /q , q * ) is the dual space of d (v, q) when v(n) is a nonincreasing sequence and satisfies the regularity condition By Hardy's inequality [13], the space l p (N) (i.e. space of all sequences (λ n ) n≥1 such that ( ∞ n=1 |λ k | p ) 1 p < ∞) is continuously embedded into ces p (N) for 1 < p < ∞ with strict embedding. In other words λ k ∈ ces p (N) if and only if the operator T(λ) = ( 1 m m k=1 |λ k |) belongs to l p (N). Inspired by the development in the continuous case, it is proved in [2,Theorem 4 Inequality (4) can also be interpreted as inclusions between the space of sequences l p v (N) and the Cesàro space of sequences ces p u (N). In fact this inequality (5) implies that the spaces l p v (N) with weight are embedded into ces p u (N) with strict embedding; as a consequence, we have that We say that the function η : N→ R belongs to the space l p λ (N) with a nonnegative weight The inclusion interpretation for the discrete spaces has been considered by Bennett in his memorial in [8]. He proved that the validation of inequality (4) is equivalent to an inclusion theorem between the spaces l p (N) and ces p (N). Precisely Bennett [8] was concerned with the multipliers from l p (N) into ces p (N), namely those sequences z = {z n } with the property that y · z ∈ ces p (N) whenever y n ∈ l p (N) and z n ∈ G(N) such that n m=1 |z m | p * = O(n). The set G(N) of all such multipliers clearly satisfies l p (N) · G(N) ⊆ ces p (N).
Since the discovery of this new way of looking at inequalities, several mathematicians such as Johnson and Mohapatra (see [1, 17-19, 21, 23, 25]) studied the generalizations of the sequence spaces l p (N) and ces p (N).
In the case of functions, the same factorization results as well as the dual space of Cesàro space are only mentioned in Bennett [8] for the unweighted spaces. A factorization result for the unweighted Cesàro function spaces was proved in Astashkin and Maligranda [5]. In [10] Carton and Heining proved a factorization result which can be considered as a weighted integral analogue of the result obtained by Bennett in [8] for the discrete Hardy operator in the unweighted case. In [7] Barza et al. extended the results proved in [5] and proved some factorization theorems for the Cesàro and Copson functions spaces with weights.
However, the results of Bennett have a big impact in many parts of analysis, but it seems that the corresponding results for weighted spaces are less studied. In some special cases it is possible to translate or adapt almost straightforward the objects and results from the continuous setting to the discrete setting or vice versa; however, in some other cases that is far from being trivial.
In this paper, we develop a new technique to study the structure of weighted Cesàro and Copson sequences spaces and prove some factorization theorems. We mention here that our technique, which is based on some useful lemma proved and designed for this purpose, can be considered as the modification of the technique used in [5] to prove the unweighted results and the technique used in [7] to prove the weighted results for the Cesàro functions space. To the best of the authors' knowledge, the results in this paper for the Cesàro sequences space have not been considered before.
We denote by H the Cesàro operator and by M the Copson operator, which are defined by Throughout the paper, the letters A, B, C, D are used for nonnegative constants independent of the relevant variables that may change from one occurrence to another. In [2] (see also [9]) the authors proved that the discrete inequality holds for all nonnegative sequence η and 1 < p < ∞ if This result proves that the operator H is bounded on the weighted space l p λ (N) if (7) holds. Also in [9] the authors proved that the inequality of Copson type holds for all nonnegative sequence η and 1 < p < ∞ if Inequality (8) is a generalization of the discrete inequality due to Copson, see [11,12]. The rest of the paper is divided into three sections: Sect. 2 is devoted to some basic lemmas that will be needed in the proofs of the main results. Section 3 is devoted to the proof of the discrete weighted Cesàro space Ces p λ (N) for p > 1, and the case when p = 1, which has not been considered before, is treated separately. In fact we are concerned with the multipliers from l p λ (N) into Ces p λ (N), namely those sequences β with the property that , which will be defined later in the next section, of all such multipliers clearly satisfies Section 4 is devoted to the same problem but for the discrete weighted Copson space Cop p λ (N). The results show that the boundedness of the discrete operators Hη and Mη on the weighted space l p λ (N) can be obtained from general norm discrete inequalities.

Basic lemmas
In this section, we state and prove the main basic lemmas that are needed in the rest of the papers. The first two are adapted from [9]. (10) In what follows, it will be convenient to use the convention b m=a y(m) = 0, whenever a > b, The following lemmas will be also needed in the proofs.
Proof By defining F(n) = n k=1 φ(k) and G(n) = n k=1 ϕ(k), we see from (12) on the left-hand side of (13), we get By noting that F(1) = H(N + 1) = 0 and making use of Applying summation by parts on the right-hand side of inequality (15), we have that Since G(1) = H(N) = 0, we have from the last inequality that Combining (15) and (16), we get which is the desired inequality (13). The proof is complete. Proof Applying summation by parts twice.

Lemma 2.5 Let ϕ, ψ be nonnegative sequences, then
Proof By defining (k) = N-1 s=k ψ(s) and applying summation by parts on the left-hand side of (17) with η(k) = (k) and v(k) = ϕ(k), we get The proof is complete.

Weighted Cesàro sequences space
In this section, we prove a factorization theorem of the discrete weighted Cesàro space Ces p λ (N); as a consequence, we recover some best known forms of the discrete Hardy type inequalities as special cases. We start by presenting the basic definitions.
with the norm We denote The infimum is taken over all possible decompositions of = η × β with η ∈ l p λ (N) and β ∈ A q λ (1-q) (N).

Definition 3.3
We denote by M * the discrete class of weights λ such that, for all n ≥ 1, it follows that for p > 1, where 1/p + 1/q = 1 and A * is the smallest positive constant such that (21) holds.

Definition 3.4
We denote by M * the discrete class of weights such that, for all n ≥ 1, it follows that the reverse of inequality (21) holds, and we denote by A * the largest positive constant for which this reverse holds.
Proof We start by proving the imbedding → i.e. we prove that the sequence = η × β ∈ Ces p λ (N), where η ∈ l p λ (N) and β ∈ A q (λ 1-q ) (N). By defining w(n) = ( n m=1 λ 1-q (m)) -1/pq and employing discrete Hölder's inequality with 1/p + 1/q = 1, we get that Now, Definition (19) Dividing (24) by n > 0 and summing from 1 to ∞ and then using (25) and applying Lemma 2.5, we get that By employing (10) with φ(x) = λ 1-q (x) and γ = 1/q < 1 and using the definition of w(n), we obtain that . By using the definition of A * and employing (11) with γ = 1/p < 1 and φ = λ(n)/n p , we get that This implies that ∈ Ces p λ (N) and and the infimum is determined over all possible factorization of . This leads to the completeness of the first part of the proof. For the reversed direction ← , we prove that Ces and let η(n) = (n) 1/p v 1/p (n), and β(n) = (n) 1/q v -1/p (n).

This implies that
which gives the left-hand side of inequality (22), and the infimum is taken over all possible decompositions of = η ·β such that η ∈ l p λ (N) and β ∈ A q (λ 1-q ) (N). The proof is complete.

Moreover,
where A * and A * are defined as above.
Now, we consider the case when p = 1, which is an independent and important case in its own. In this case, we consider the space Ces 1 λ (N) which will be obtained from Ces p λ (N) by putting p = 1. On the other hand, if we consider β ∞ = sup n>0 |β(n)| < ∞, we see that This allows us to replace the space A 1 λ (N) with the space l ∞ (N) with a norm β ∞ = ess sup n>0 |β(n)| < ∞. Now, we consider a new space U of weights which is defined by and assume that there exists a positive constant P * which is the smallest constant such that the inequality holds, and there exists a positive constant P * which is the largest constant for which the reverse of (30) holds. In the following, we denote where the infimum is taken over all possible decompositions of = η × β with η ∈ l 1 λ (N) and β ∈ l ∞ (N).
Then from (40) and (41)  which leads to the proof of the left-hand side of (33). The proof is complete.

Weighted Copson sequences space
In this section, we prove a theorem of factorization of the Copson space Cop p λ (N) of discrete weights; as a consequence, we obtain the well-known forms of the discrete Copson type inequalities with best constants. We consider the two cases p > 1 and p = 1.