A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space

Let $M$ be the Doob maximal operator on a filtered measure space and let $v$ be an $A_p$ weight with $1<p<+\infty$. We try proving that \begin{equation}\lVert M f\rVert _{L ^{p}(v) }\leq p^{\prime}[v]^{\frac{1}{p-1}}_{A_p}\lVert f\rVert _{L ^{p} (v)},\end{equation} where $1/p+1/p^{\prime}=1.$ Although we do not find an approach which gives the constant $p^{\prime},$ we obtain that \begin{equation}\lVert M f\rVert _{L ^{p}(v) }\leq p^{\frac{1}{p-1}}p^{\prime}[v]^{\frac{1}{p-1}}_{A_p}\lVert f\rVert _{L ^{p} (v)}, \end{equation} with $\lim\limits_{p\rightarrow+\infty}p^{\frac{1}{p-1}}=1.$

where 1/p + 1/p ′ = 1. Although we do not find an approach which gives the constant p ′ , we obtain that

Introduction
Let M be the Doob maximal operator on a filtered measure space. For 1 < p < +∞, it is well known (see e.g. [10]) that where 1/p + 1/p ′ = 1 and p ′ is the best constant. Let v be an A p weight with 1 < p < +∞. Tanaka and Terasawa [14] proved that For a Euclidean space with a dyadic filtration, the dyadic maximal operator is the above Doob maximal operator. For the dyadic maximal operator, the constant 1/(p − 1) is the optimal power on [v] Ap (see e.g. [11] or [9]) . It follows that the constant 1/(p − 1) is also the optimal power on [v] Ap for the Doob maximal operator M.
In this note, we estimate the constant C in (1.2). Substituting v = 1 into (1.2), we get (1.1). Thus, we conjecture that the constant C equals p ′ in (1.2). But we do not find an approach which gives the constant C = p ′ . Our results are as follows.

Theorem 1.3.
Let v be a weight and 1 < p < ∞. We have the inequality  (2) Using the construction of principal sets [14] and the conditional sparsity [3], we have C = a 2 η (p ′ −1) p ′ , where a, η are the constants in the construction of principal sets (Appendix A).
Cao and Xue [1] (see also the references therein) used the atomic decomposition to study weighted theory on the Euclidean space, but we do not know whether it is possible on the filtered measure space. This paper is organized as follows. Sect. 2 consists of the preliminaries for this paper. In Sect. 3 we give the proof of Theorem 1.3 , and in Sect. 4 we compare p 1 p−1 with a 2 η (p ′ −1) . In order to keep track the constants in our paper, we modify the construction of principal sets in Appendix A.

Preliminaries
The filtered measure space was discussed in [6,14], which is abstract and contains several kinds of spaces. For example, a doubling metric space with systems of dyadic cubes was introduced in Hytönen and Kairema [4]. In order to develop discrete martingale theory, a probability space endowed with a family of σ-algebra was considered in Long [10]. In addition, a Euclidean space with several adjacent systems of dyadic cubes was mentioned in Hytönen [7]. Because the filtered measure space is abstract, it is possible to study these spaces together( [5,12,13]). As is well known, Lacey, Petermichl and Reguera [8] studied the shift operators, which is related to the martingale theory on a filtered measure space. When Hytönen [7] solved the conjecture of A 2 , those operators are very useful.
2.1. Filtered Measure Space. Let (Ω, F , µ) be a measure space and let F 0 = {E : E ∈ F , µ(E) < +∞}. As for σ-finite, we mean that Ω is a union of (E i ) i∈Z ⊂ F 0 . We only consider σ-finite measure space (Ω, F , µ) in this paper. Let B be a sub-family of F 0 and let f : Ω → R be measurable on (Ω, F , µ). If for all B ∈ B, we have B |f|dµ < +∞, then we say that f is B-integrable. The family of the above functions is denote by L 1 B (F , µ). Let B ⊂ F be a sub-σ-algebra and let f ∈ L 1 B 0 (F , µ). Because of σ-finiteness of (Ω, B, µ) and Radon-Nikodým's theorem, there is a unique function denoted by E(f|B) ∈ Letting (Ω, F , µ) with a family (F i ) i∈Z of sub-σ-algebras satisfying that (F i ) i∈Z is increasing, we say that F has a filtration (F i ) i∈Z . Then, a quadruplet (Ω, F , µ; (F i ) i∈Z ) is said to be a filtered measure space. It is clear that L 1 then τ is said to be a stopping time. We denote the family of all stopping times by T . For i ∈ Z, we denote T i := {τ ∈ T : τ ≥ i}.

Operators and Weights.
Let f ∈ L. The Doob maximal operator is defined by For i ∈ Z, we define the tailed Doob maximal operator by For ω ∈ L with ω ≥ 0, we say that ω is a weight. The set of all weights is denoted by L + . Let B ∈ F , ω ∈ L + . Then Ω χ B dµ and Ω χ B ωdµ are denoted by |B| and |B| ω , respectively. Now we give the definition of A p weights. Definition 2.1. Let 1 < p < ∞ and let ω be a weight. We say that the weight ω is an A p weight, if there exists a positive constant C such that

Approaches of Theorem 1.3
Proof. We prove that (1.4) implies (1.5). For i ∈ Z and B ∈ F 0 i , we let f = χ B . Then In order to prove (1.6), we provide the three approaches which we mentioned in Remark 1.7.
Approach (1). It is clear that Then we have Using the boundedness of Doob maximal operators M v and M σ , we obtain Approach (2). For i ∈ Z, k ∈ Z and Ω 0 ∈ F 0 i , we denote We claim that where a, η are the constants in the construction of principal sets (Appendix A). To see this, denote h = fσχ P 0 . For the above i, P 0 and h, we construct principal sets. Then, Lemma A.3 shows that (3.10) To estimate |E(P)| v . For the sake of simplicity, we denote E F K 1 (P) (·) by E P (·) without confusion. We now estimate |E(P)| v as follows: In the view of the definition of A p and the construction of P, we have Noting that the conditional expectation satisfies Hölder's inequality, we have Because E(P) is a subset of P and a K 2 (P)−1 χ P ≤ E P (h)χ P , we obtain that It follows from (3.10) and the boundedness of Doob maximal operator M σ that which implies (3.9). Furthermore, Ap Ω 0 f p σdµ.
Noting that (Ω, F , µ) is a σ-finite measure space, we obtain that Ap Ω f p σdµ 1 p .

Then we denote
It follows that A k,j ∈ F τ k , B k,j ⊆ A k,j . It is clear that {B k,j } k,j is a family of disjoint sets and Following from Applying the A p condition Letting X := Z 2 and we have that ϑ is a measure on X. For f ∈ L p (vdµ) and λ > 0, we denote It follows that In view of the boundedness of Doob maximal operator M v , we get that Therefore Ap Ω M σ (fσ −1 ) p σdµ.
Using the boundedness of Doob maximal operator M σ , we conclude that (3.30) Ap Ω |f| p vdµ.
(2) We claim that the function ψ(p) is decreasing on (1, +∞). It suffices to show that ψ ′ (p) < 0. We have It is clear that ψ ′ (p) < 0 if and only if 1 − 1 p + ln 1 p < 0. Let s(t) = 1 − t + ln t with t ∈ (0, 1]. Because of s ′ (t) = 1 t − 1 > 0 on (0, 1), the function s(t) is strictly increasing on (0, 1]. It follows from s(1) = 0 that s(t) < 0 on (0, 1). That is 1 − 1 p + ln 1 p < 0 with p > 1. Thus ψ(p) is decreasing on (1, +∞). At the end of Section 4, we check our work with graphing device in Figure 1. The construction of principal sets first appeared in Tanaka and Terasawa [14], and Chen, Zhu, Zuo and Jiao [2,3] found the conditional sparsity of the construction, which is new and useful. We will use the construction of principal sets. Because we keep track the constants of the conditional sparsity, we will give the modifications in the construction of principal sets in this Appendix.
For i ∈ Z, h ∈ L + , a > 1 and k ∈ Z, stopping times are defined by Let where Ω 0 ∈ F 0 i , then P 0 ∈ F 0 i . We denote K 1 (P 0 ) := i and K 2 (P 0 ) := k. Then we define P 1 := {P 0 }, which is the first generation P 1 . Now we show how to define the second one. Let τ P 0 := τχ P 0 + ∞χ P c 0 , where P c 0 = Ω \ P 0 . Let P be a subset of P 0 with µ(P) > 0. If there is i < j and k + 1 < j such that P = {a l−1 < E(h|F j ) ≤ a l } ∩ {τ P 0 = j} ∩ P 0 = {a l−1 < E(h|F j ) ≤ a l } ∩ {τ = j} ∩ P 0 , we say that P is a principal set of P 0 . We denote K 1 (P) := j and K 2 (P) := l. Letting P(P 0 ) be the family of the above principal sets of P 0 , we say that P 2 := P(P 0 ) is the second generation.
where η = a/(a − 1). Now, we represent the tailed Doob maximal operator by the principal sets, which is the following lemma. Lemma A.3. Let h ∈ L + , a > 1 and i ∈ Z. For k ∈ Z and Ω 0 ∈ F 0 i , we let P 0 := {a k−1 < E(h|F i ) ≤ a k } ∩ Ω 0 .