Embeddings between weighted Tandori and Ces\`{a}ro Function Spaces

We give the characterization of the embeddings between weighted Tandori and Cesàro function spaces using the combination of duality arguments for weighted Lebesgue spaces and weighted Tandori spaces with estimates for the iterated integral operators.


Introduction
Given two function spaces X, Y and an operator T, a standard problem is characterizing the conditions for which T maps X into Y . If X and Y are (quasi) Banach spaces of measurable functions, a bounded operator T : X → Y satisfies the inequality T f Y ≤ c f X for all f ∈ X where c ∈ (0, ∞). When T is the identity operator I, we say that X is embedded into Y and write X ֒→ Y . The least constant c in the embedding X ֒→ Y is I X→Y .
In this paper, we find the optimal constants in the embedding between weighted Tandori and Cesàro function spaces. We shall begin with the definitions of the function spaces considered in this paper.
Given a measurable function f on E, we set We do not aim to give a thorough set of references on the history of these spaces. Instead, we refer the reader to survey paper by Astashkin and Maligranda [1] and the references therein, where the comprehensive exposition of history on the structure of Cesàro and Copson function spaces are given. In this paper our primary focus is the following inequality for all measurable functions where 0 < p i , q i ≤ ∞, i = 1, 2. There is more than one motivation to study inclusion between Cesàro and Copson spaces. First of all when p 1 = q 1 or p 2 = q 2 , weighted Cesàro and Copson function spaces coincide with some weighted Lebesgue spaces (see [5,), thus inequality (1.1) is a generalization of the well-known weighted direct and reverse Hardy-type inequalities (e.g. [10], [4], [14]). Another justification is to give the characterization of pointwise multipliers between two spaces of Cesàro and Copson type, because it reduces to the characterization the embeddings between these spaces. In [8,Section 7] Grosse-Erdmann considered the multipliers between the spaces of p-summable sequences and Cesàro and Copson sequence spaces. He also introduced corresponding function spaces but the characterization of the multipliers between two spaces of Cesàro and Copson type remained open for both sequence and function spaces for a long time.
The characterization of the inequality (1.1) is given in one parameter case when p 1 = p 2 = 1, [2]. Moreover, it was shown that the inequality is reversed when 0 < p < 1. In [3], inequality (1.1) is considered for two different parameters in the special case p 1 = p 2 = 1, under the restriction q ≥ 1 in order to characterize the embeddings between some Lorentz-type spaces. Recently, in [5] the two sided estimates for the best constant in (1.1) is given for four different parameters 0 < p 1 , p 2 , q 1 , q 2 < ∞ under the restriction p 2 ≤ q 2 , moreover, using these results, in [7] pointwise multipliers between Cesàro and Copson function spaces is given for some ranges of parameters.
Furthermore, in 2015, Lesnik and Maligranda ( [11], [12]) began studying these spaces within an abstract framework, where they replaced the role of weighted Lebesgue spaces with a more general function space X. For a Banach space X, they defined Cesàro space CX, Copson space C * X and Tandori space X as the set of all measurable functions, respectively, with the following norms: In [13], they named X as the generalized Tandori spaces in honour of Tandori who provided dual spaces to the spaces CL ∞ [0, 1] in [15]. Their definition is related to our definition in the following way: CL p,w = Ces 1,p (x −1 w(x), 1), C * L p,w = Cop 1,p (w, x −1 ), L p,w = Cop ∞,p (w, 1).
We should note that recently in [9] multipliers between CL p and CL q are given when 1 < q ≤ p ≤ ∞. We want to continue this research. In this paper, we will consider the embeddings between weighted Tandori and weighted Cesàro function spaces, namely, we will give the characterization of the following inequality, for all measurable functions where p, q, r ∈ (0, ∞) with p < q. The restriction on the parameters arises from the duality argument. The key ingredient of the proof is combining characterizations of the associate spaces of Tandori spaces, namely, the reverse Hardy-type inequality for supremal operators which was given in [14] with the characterizations of some iterated Hardy-type inequalities. Throughout the paper, expressions of the form 0 · ∞ or 0 0 are taken as zero. For p ∈ (1, ∞), we define p ′ = p p−1 . We write A ≈ B if there exist positive constants α, β independent of relevant quantities appearing in expressions A and B such that The symbol M will stand for the set of all measurable functions on (0, ∞), and we denote the class of non-negative elements of M by M + .
We sometimes omit the differential element dx to make the formulas simpler when the expressions are too long.
The paper is structured as follows. In Section 2, we formulate the main results of this paper. In Section 3, we collect some properties and necessary background material. Finally, in the last section, we give the proofs of our main results.

Main Results
It is convenient to start this section by recalling some properties of the weighted Cesàro and Copson spaces. Let 0 < p, q ≤ ∞. Assume that u is a non-negative measurable function and v is a weight. We will always assume that u q,(t,∞) < ∞ for all t > 0 and u q,(0,t) < ∞ for all t > 0, when considering weighted Cesàro and Copson function spaces, respectively. Otherwise, these spaces consist only of functions equivalent to zero (see, [5, Lemmas 3.1-3.2]).
In this section, we will formulate the least constant in the embedding holds. Therefore, it is enough to consider the three weighted case (2.1).
Remark 2.2. Note that, when p = q or r = ∞, this problem is not interesting since it reduces to the characterizations of Hardy-type inequalities and can be found in [5], therefore we will consider the cases when r < ∞. On the other hand, we have the restriction p < q, which arises from the duality argument.

Background Material
In this section we quote some known results. Let us start with the characterization of the reverse Hardy-type inequality for supremal operator, that is, for all non-negative measurable functions f on (0, ∞) where 0 < p, q < ∞.
Moreover, the least possible constant C in (3.1) satifies C ≈ A 1 .
(ii) If p < q, then inequality (3.1) holds for all non-negative measurable functions f on (0, ∞) if and only if A 2 < ∞ and A 3 < ∞, where and Moreover, the least possible constant C in (3.1) satifies C ≈ A 2 + A 3 .
We next recall the characterization of the weighted iterated inequality involving Hardy and Copson operators, that is, (3.5) Note that the characterization of inequality (3.5) is given in [6]. In the next theorem, we provide a modified version of [6, Theorem 3.1], using the gluing lemmas presented in the recent paper [7]. Denote by Theorem 3.2. Let 1 < p < ∞ and 0 < q < ∞. Assume that u ∈ M + and v, w ∈ M + such that • v(t) > 0, V(t) < ∞ for all t ∈ (0, ∞) and lim t→∞ V(t) = ∞, • ∞ t w(s) q ds < ∞ for all t ∈ (0, ∞) and Moreover, the least possible constant C in (3.5) satifies C ≈ B 1 + B 2 .
(ii) If q < p, then (3.5) holds for all non-negative measurable functions f on (0, ∞) if and only if B 3 < ∞ and B 4 < ∞, where Moreover, the least possible constant C in (3.5) satifies C ≈ B 3 + B 4 .
Proof. The proof is the combination of [6, Theorem 3.
Observe that, interchanging supremum gives Thus, the problem reduced to the characterization of reverse Hardy-type inequalities for supremal operator. It remains to apply [Theorem 3.1, (i)] when r ≤ p and [Theorem 3.1, (ii)] when p < r.
Proof of Theorem 2.4 Let 0 < r ≤ p < q < ∞. We have Since q/p ∈ (1, ∞), by the duality in weighted Lebesgue spaces, we have Interchanging supremum and Fubini's Theorem gives that where, and R(g) is the best constant in the inequality for every fixed g ∈ M + , Now, we can apply Theorem 3.1 by taking the parameters p, r, and weights Since r ≤ p, according to the first case in Theorem 3.1, Interchanging suprema yields that From Fubini's Theorem and duality in weighted Lebesgue spaces with q/p ∈ (1, ∞) again, it follows that Observe that, Thus we arrive at C ≈ I 5 + I 6 .
Proof of Theorem 2.5 Let 0 < p < r < ∞ and 0 < p < q < ∞. Using the steps identical to the preceding proof, which relies on q/p ∈ (1, ∞), duality in weighted Lebesgue spaces, and Fubini's Theorem one can see that (4.1) holds. Since p < r, applying the second case of Theorem 3.1, we obtain that R(g) ≈ First observe that, using duality principle again we have It remains to apply Theorem 3.2. To this end we should again split this case into two parts.
(i) If r ≤ q, then applying the first case in Theorem 3.2, we obtain that C 1 ≈ I 8 + I 9 and the result follows.
(ii) If q < r, then applying the second case in Theorem 3.2, we obtain that C 1 ≈ I 10 + I 11 and the result follows.