Abstract
Let \(0<q\le \infty \), b be a slowly varying function and \( \Phi : [0,\infty ) \longrightarrow [0,\infty ) \) be an increasing function with \(\Phi (0)=0\) and \(\lim \limits _{r \rightarrow \infty }\Phi (r)=\infty \). In this paper, we introduce a new class of function spaces \(L_{\Phi ,q,b}\) which unify and generalize the Lorentz-Karamata spaces with \(\Phi (t)=t^p\) and the Orlicz-Lorentz spaces with \(b\equiv 1\). Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob’s maximal inequalities by using Hardy’s inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38, 39] and the condition \(q_{\Phi ^{-1}}<1/q\) in [24], respectively.
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1 Introduction
Martingale theory is widely studied in the field of mathematical physics, harmonic analysis and probability theory. The study of martingale theory can be traced back to Doob [11] and Burkholder and Gundy [8] who established the well-known inequality named after them. Since then, several significant results had been developed such as the duality between the martingale space and BMO and the atomic decompositions. The classical martingale theory is built on Lebesgue space \(L_p\) (see [19, 59, 60]). Now the theory had been extended to other important function spaces in analysis.
Another interesting topic we shall touch in our work is Lorentz-Karamata space \(L_{p,q,b}\). These spaces are defined by means of a slowly varying function b and generalized Lebesgue spaces, Lorentz spaces, Lorentz-Zygmund spaces and even generalized Lorentz-Zygmund spaces. They were introduced by Edmunds et al. [13] in 2000 and studied by Neves [48], Gogatishvili et al. [20]. This class offers not only a more general and unified insight for these families of spaces but also provides a framework in which it is easier to appreciate the central issues of different results, see [12, 17, 21, 30] and the references therein.
In probability theory, the study of Lorentz-Karamata martingale spaces has attracted more and more attention in martingale theory. By using Karamata theory, Ho [28] combined the martingale theory with Lorentz-Karamata spaces. He introduced the Lorentz-Karamata Hardy martingale spaces and discussed the atomic decompositions, dual theorems and some martingale inequalities of these spaces. Motivated by [37], Jiao et al. [38] further studied the Lorentz-Karamata Hardy martingale spaces and partially extended the results of [28] to a wider range. Inspired by [28, 38], Liu et al. [40] investigated the Lorentz-Karamata Hardy spaces for Banach space-valued martingales; Zhou et al. [69] and Wu et al. [64] studied the weak Orlicz-Karamata-Hardy martingale spaces and Orlicz-Karamata martingale spaces, respectively. In 2017, Liu and Zhou [39] investigated the boundedness of fractional integral operators on the Lorentz-Karamata martingale spaces.
In this paper, we introduce a class of Orlicz-Lorentz-Karamata spaces, which are much more wider than the Lorentz-Karamata spaces or the Orlicz-Lorentz spaces and other important function spaces, and then develop a theory of the martingale Hardy spaces in the new framework. The results obtained in this paper greatly broaden the scope of the previous results in Lorentz-Karamata Hardy martingale spaces and Orlicz-Lorentz Hardy martingale spaces. Besides the introduction, the paper is organized in eight sections as follows.
Section 2 gives the fundamental properties of the Orlicz-Lorentz-Karamata spaces. These properties are very useful to prove later the atomic characterization of the new Hardy spaces. Based on these Orlicz-Lorentz-Karamata spaces, we introduce a new kind of Hardy spaces, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces, which generalize the Lorentz-Karamata Hardy martingale spaces as well as the Orlicz-Lorentz Hardy martingale spaces introduced very recently by Ho [28] and Hao and Li [24].
Section 3 is devoted to Doob’s maximal inequalities on Orlicz-Lorentz-Karamata spaces. In [25], the authors presented Doob’s inequality on Orlicz-Lorentz-Karamata spaces under the following conditions: \(\Phi \) is an Orlicz function with \(p_{\Phi }>1\), \(1\le q\le \infty \) and b is a slowly varying function. Here \(p_{\Phi }\) and \(q_\Phi \) is the lower and upper Simonenko index defined as
and
However, Doob’s inequality on Orlicz-Lorentz-Karamata spaces for \(0<q< 1\) was still an open problem. We solve this problem in Theroem 3 by using a new characterization of Hardy-type inequality. More exactly, we generalize Doob’s maximal inequality on Orlicz-Lorentz-Karamata spaces for \(0< q\le \infty \) and for \(\Phi \) being an Orlicz function with lower type \(p>1\) (see Section 2 for the definition of lower type). Note that the condition of lower type is weaker than the lower Simonenko index.
Section 4 presents the atomic decompositions of the five Orlicz-Lorentz-Karamata Hardy martingale spaces. Roughly speaking, under some weak conditions, we prove that a martingale \(f=(f_n)_{ n \ge 0 }\in H_{\Phi ,q,b}^s\) (resp. \(\mathcal {Q}_{\Phi ,q,b}\), \(\mathcal {P}_{\Phi ,q,b}\)) if and only if there exist a sequence \((a^k)_{ k \in \mathbb {Z} } \) of \((1,\Phi ,\infty )\) (resp. \((2,\Phi ,\infty )\), \((3,\Phi ,\infty )\)) atoms and a sequence \((\mu _k)_{k\in \mathbb {Z}}\) of real numbers
(\(\kappa \) is a positive constant and \(\tau _k\) is the stopping time associated with \( a^k \)) such that
and
where the infimum is taken over all decompositions of f as above. When the stochastic basis is regular, we also establish atomic decompositions for the martingale Hardy spaces \(H_{\Phi ,q,b}^M\) and \(H_{\Phi ,q,b}^S\). As corollaries of our theorems in this section, we generalize several known theorems from Lorentz-Karamata Hardy spaces and Orlicz-Lorentz Hardy spaces to Orlicz-Lorentz-Karamata Hardy spaces. Amongst others, we remove the restricting condition \(0< q\le p\le 1\) of Theorem 5.4 in [28]; the restricting condition that b is non-decreasing in Theorem 3.1 of [38]; the restricting conditions \(0<q\le 1\) and \(q_{\Phi ^{-1}}<1/q\) of Theorems 3.1 and 3.2 in [24].
Section 5 shows the fundamental martingale inequalities and the relation of the different Orlicz-Lorentz-Karamata Hardy martingale spaces. The method we used here is to establish a sufficient condition for the \(\sigma \)-sublinear operator to be bounded from martingale Hardy spaces to Orlicz-Lorentz-Karamata spaces.
Section 6 applies the atomic decompositions of Orlicz-Lorentz-Karamata Hardy martingale spaces above to obtain that if \(0<q\le 1\), then the dual space of \(H_{\Phi ,q,b}^s\) is the generalized \(BMO_2\) (usually denoted by BMO) martingale space \(BMO_{2,\Phi ,b}\); if \(1<q<\infty \), then the dual space of \(H_{\Phi ,q,b}^s\) is \(BMO_{2,q,\Phi ,b}\). The conclusions improve the recent results in [27, 37]. As a consequence, Theorems 1.2 and 1.5 in [38] are also true without the assumption that the slowly varying function b is non-decreasing. Moreover, we remove the restricting condition \(q_{\Phi ^{-1}}<1/q\) of Theorem 4.3 in [24].
Section 7 describes the John-Nirenberg inequality for the martingale spaces \(BMO_{r,q,\Phi ,b}\). Based mainly on the duality and the John-Nirenberg inequality, the space BMO plays an important role in classical analysis and martingale theory, see [38, 56, 60] and the references therein. The main result of this section (see Theorem 13) completes and generalizes the recent conclusion from Jiao et al. [37] and [38].
The last part of the paper, Section 8, extends the boundedness of the fractional integrals to a much more general setting. The fractional integrals are useful tools to analyze the function spaces in harmonic analysis, see [9, 14, 32, 52]. In probability theory, the fractional integrals were first introduced by Chao and Ombe [10] for dyadic martingales. The fractional integrals as special cases of martingale transforms (see [7]) have recently attracted widely attention. Now the fractional integrals have been defined for more general martingales than in [2], [23], [37] and [46, 47]. The goal of this section is to study the boundedness of the fractional integrals for Orlicz-Lorentz-Karamata Hardy martingale spaces. The results are very interesting and improve and generalize several known results, amongst others Theorem 4.4 in [39].
At the end of this section, we make some conventions. Throughout this paper, we denote by C the absolute positive constant that is independent of the main parameters involved, but whose value may differ from line to line. The symbol \(f \preceq g\) stands for the inequality \(f \le C g\). When we write \(f \approx g\), this stands for \( f \preceq g \preceq f \).
2 Orlicz-Lorentz-Karamata spaces
In this section, we give some preliminaries necessary for the whole paper.
2.1 Orlicz spaces
Let us first recall some basic properties on Orlicz functions and Orlicz spaces in this subsection. Let \((\Omega ,\mathcal {F},\mathbb {P})\) be a complete probability space and f be an \(\mathcal {F}\)-measurable function defined on \(\Omega \). The distribution function of f is the function \(\lambda _s(f)\) defined by
Denote by \(f^*\) the decreasing rearrangement of f, defined by
with the convention that \(\inf \emptyset =\infty \).
Recall that an Orlicz function \(\Phi \) is an increasing function such that \(\Phi (0)=0\) and \(\lim \limits _{r \rightarrow \infty }\Phi (r)=\infty \). The Orlicz space \(L_\Phi =L_\Phi (\Omega ,\mathcal {F},\mathbb {P})\) is the set of all \(\mathcal {F}\)-measurable functions f satisfying \(\mathbb {E}\big (\Phi (c|f|)\big ) < \infty \) for some \(c > 0\) and
where \(\mathbb {E}\) denotes the expectation operator or the integral with respect to \(\mathbb {P}\).
Remark 1
-
(1)
If \(\Phi (t)=t^p \;(0<p<\infty )\), then \(L_{\Phi }\) is the Lebesgue space \(L_p\). In this case we denote \(\Vert \cdot \Vert _{L_\Phi }\) by \(\Vert \cdot \Vert _{L_p}\).
-
(2)
If \( \Phi \) is a convex Orlicz function, then the functional \(\Vert \cdot \Vert _{L_\Phi }\) is a norm and thereby \(L_\Phi \) is a Banach space.
-
(3)
By a simple calculation, one can check that for any \(A\in \mathcal {F}\), \(\mathbb {P}(A)>0\),
$$\begin{aligned} \Vert \chi _A\Vert _{L_\Phi } = \frac{1}{\Phi ^{-1}(\frac{1}{\mathbb {P}(A)})}, \end{aligned}$$where \(\Phi ^{-1}\) denotes the left inverse function of \(\Phi \), i.e. \(\Phi ^{-1}(t)=\inf \{u\ge 0: \Phi (u)>t\}\).
Definition 1
Let \(0<p<\infty \). One says that \(\Phi \) is lower (resp. upper) type p if there exists a positive constant \(C_{p}\) depending on p such that, for any \(t \in [0,\infty )\) and \(s \in (0, 1)\) (resp. \(s \in [1, \infty )\)),
Example 1
If \(\Phi (t) = t^p\) for all \(t \in (0,\infty )\) and \(p \in (0, \infty )\), then the lower and upper types of \(\Phi \) are equal to p.
For \(0< p< r < \infty \), let
Then \(\Phi \) is of lower type p and upper type r.
For \(p\in (0,\infty )\) and \(r\in \mathbb {R}\), let
where \(c_{p,r}\) is a positive constant that ensures the increasing property of \(\Phi \) on \((0,\infty )\). Then the lower and upper types of \(\Phi \) are equal to p.
For \(0< p< r < \infty \), consider the increasing function
Then \(\Phi \) is of lower type \(r-p\) and upper type r. The last two examples can be found in [15].
For a function \(\varphi :[0,\infty )\rightarrow (0,\infty )\), if there is a non-decreasing (resp. non-increasing) function \(\phi \) such that \(\varphi \approx \phi \), then we say that \(\varphi \) is equivalent to a non-decreasing (resp. non-increasing) function on \([0,\infty )\). By the definitions of lower and upper type, it is easy to see the following two lemmas.
Lemma 1
Let \(\Phi \) be an Orlicz function of lower type \(p \in (0,\infty )\). Then we have
-
(i)
the function \(t \longmapsto t^{-p} \Phi (t)\) is equivalent to a non-decreasing function;
-
(ii)
the function \(t \longmapsto t^{-1/{p}} \Phi ^{-1}(t)\) is equivalent to a non-increasing function;
-
(iii)
\(\Phi ^{-1}(2t)\le C_p\Phi ^{-1}(t)\) for any \(t\in [0,\infty )\), where \(C_p\) is depending only on p.
Lemma 2
Let \(\Phi \) be an Orlicz function of upper type \(p \in (0,\infty )\). Then we have
-
(iv)
the function \(t \longmapsto t^{-p} \Phi (t)\) is equivalent to a non-increasing function;
-
(v)
the function \(t \longmapsto t^{-1/{p}} \Phi ^{-1}(t)\) is equivalent to a non-decreasing function;
-
(vi)
\(\Phi ^{-1}(2t)\ge C_p\Phi ^{-1}(t)\) for any \(t\in [0,\infty )\), where \(C_p\) is depending only on p.
The following properties of Orlicz spaces will be used in the sequel.
Lemma 3
([68]) Let \(\Phi \) be an Orlicz function with lower type \(p\in (0,\infty )\). Then \(\Vert \cdot \Vert _{L_\Phi }\) is a quasi-norm.
Remark 2
Let \(\Phi \) be an Orlicz function with lower type \(p\in (0,\infty )\). It follows from Lemma 3 and the Aoki-Rolewicz theorem (see [1]) that there exists \(\varrho \in (0,1]\) such that
for all \(h_1,h_2,\cdots ,h_n\) (for any \(n \ge 1\)) in \(L_\Phi \).
Lemma 4
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \). For \(A\in \mathcal {F}\), we have
Proof
It is sufficient to consider the case of \(\mathbb {P}(A)\ne 0\). It follows from Lemmas 1 (ii) and 2 (v) that
and
According to Remark 1 (3), we have
and
This completes the proof. \(\square \)
See [18, 43, 53] for more information on Orlicz functions and Orlicz spaces.
2.2 Orlicz-Lorentz-Karamata spaces
We recall the definition of slowly varying function in order to define the Orlicz-Lorentz-Karamata spaces.
Definition 2
A Lebesgue measurable function \(b :[ 1,\infty ) \longrightarrow (0,\infty )\) is said to be a slowly varying function if for any given \(\epsilon >0\), the function \(t^\epsilon b(t)\) is equivalent to a non-decreasing function and the function \(t^{-\epsilon }b(t)\) is equivalent to a non-increasing function on \([1,\infty )\).
The detailed study of Karamata theory, properties and examples of slowly varying functions can be found in [6, 13].
Proposition 1
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \) and b be a slowly varying function. Then the function \(b_1\) defined on \([1,\infty )\) by \(b_1(t)=b(\Phi (t))\) is a slowly varying function.
Proof
We firstly prove that for any \(\varepsilon >0\), \(t^\varepsilon b_1(t)\) is equivalent to a non-decreasing function. Indeed, for \(t\ge 1\), there is
Since \(\Phi \) is of upper type \(p_+\in (0,\infty )\), it is easily to see that \(\frac{t^{p_+}}{\Phi (t)}\) is equivalent to a non-decreasing function. Moreover, b is a slowly varying function, which implies that \(\Phi (t)^{\varepsilon /p_+}b(\Phi (t))\) is equivalent to a non-decreasing function. Hence we obtain that \(t^\varepsilon b_1(t)\) is equivalent to a non-decreasing function for any \(\varepsilon >0\).
Similarly, we can prove that \(t^{-\varepsilon } b_1(t)\) is equivalent to a non-increasing function for any \(\varepsilon >0\). Thus \(b_1\) is a slowly varying function. \(\square \)
Given a slowly varying function b on \([1,\infty )\), we denote by \(\gamma _b\) the positive function defined by
Proposition 2
[12]) Let b be a slowly varying function. For any given \(\epsilon >0\), the function \(t^\epsilon \gamma _b(t)\) is equivalent to a non-decreasing function and \(t^{-\epsilon }\gamma _b(t)\) is equivalent to a non-increasing function on (0, 1].
Based on this, we now present the definition of the Orlicz-Lorentz-Karamata spaces.
Definition 3
Let \( \Phi \) be an Orlicz function, \(0<q\le \infty \) and let b be a slowly varying function. The Orlicz-Lorentz-Karamata space, denoted by \(L_{\Phi , q, b}\), consists of those measurable functions f with \(\Vert f\Vert _{L_{\Phi , q, b}}<\infty \), where
These spaces are generalizations of classical Lorentz spaces, Lorentz-Karamata spaces and Orlicz-Lorentz spaces.
Remark 3
-
(1)
If \(b\equiv 1\), \(L_{\Phi ,q,b}\) gives to the Orlicz-Lorentz space \(L_{\Phi ,q}\) introduced and studied in [24]. The Orlicz-Lorentz space \(L_{\Phi ,q}\) coincides with the classical Lorentz space \(L_{p,q}\) when \(\Phi (t) = t^p\) for \(0<p<\infty \). If \(\Phi (t)=t^q\) for \(0<q<\infty \), then \(L_{\Phi ,q}\) are the usual Lebesgue spaces \(L_{q}\). Moreover, if \(q=\infty \), the space \(L_{\Phi ,q}\) reduces to the weak Orlicz space.
-
(2)
If \(\Phi (t) = t^p\) for \(0<p<\infty \), \(L_{\Phi ,q,b}\) becomes the Lorentz-Karamata space \(L_{p,q,b}\). In this situation, when \(b(t)=1+\log t\), the Lorentz-Karamata space reduces to the Lorentz-Zygmund space introduced and studied in [4].
-
(3)
Let \(0<p<\infty \), \(m \in \mathbb {N}\) and \(\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _m) \in \mathbb {R}^m\). Define the family of the positive functions \(\{\ell _k\}^m_{k=0}\) on \((0,\infty )\) by
$$\begin{aligned} \ell _0(t) = 1/t,\quad \ell _k(t)=1+\log \big (\ell _{k-1}(t)\big ),\quad 0<t\le 1,\;1\le k\le m. \end{aligned}$$Moreover, define
$$\begin{aligned} \Theta _\alpha ^m(t)=\prod _{k=1}^m\ell _k^{\alpha _k}(t). \end{aligned}$$The generalized Lorentz-Zygmund space consists of all \(\mathcal {F}\)-measurable functions f such that \(\Vert f\Vert _{L_{p,q,\alpha }}<\infty \) where
$$\begin{aligned} \big \Vert f\big \Vert _{L_{p,q,\alpha }}= \left\{ \begin{array}{l} \Big (\int _0^1\big (t^{1/p}\Theta _\alpha ^m(t)f^*(t)\big )^q \frac{dt}{t}\Big )^{1/q} \ \ \text {if}\;0<q<\infty ,\\ \sup \limits _{t>0} t^{1/p}\Theta _\alpha ^m(t)f^*(t) \ \quad \quad \quad \ \quad \;\; \text {if}\; q=\infty . \end{array} \right. \end{aligned}$$
It is easy to see that the generalized Lorentz-Zygmund space is a member of Lorentz-Karamata spaces (see also [12, 28]).
Lemma 5
Let \(\Phi \) be an Orlicz function, \(0<q\le \infty \) and let b be a slowly varying function. Then the following properties hold:
-
(1)
\( \Vert f\Vert _{L_{\Phi ,q,b}} \ge 0 \) and \( \Vert f\Vert _{L_{\Phi ,q,b}} = 0 \) if and only if \(f=0\);
-
(2)
\(\Vert \lambda \cdot f\Vert _{L_{\Phi ,q,b}}=|\lambda |\cdot \Vert f\Vert _{L_{\Phi ,q,b}}\) for any \(\lambda \in \mathbb {C}\);
-
(3)
For any \(A\in \mathcal {F}\) with \(\mathbb {P}(A)>0\), we have
$$\begin{aligned} \Vert \chi _A\Vert _{L_{\Phi ,q,b}} = \gamma _b\big (\mathbb {P}(A)\big )\Vert \chi _A\Vert _{L_\Phi }. \end{aligned}$$(2.1)
Proof
The properties (1) and (2) can be easily checked by direct calculation. For (3), it follows from the definition of \(\Vert \cdot \Vert _{L_{\Phi ,q,b}}\) that
where \(0<q<\infty \). The case of \(q=\infty \) can be proved analogously. \(\square \)
Lemma 6
Let \(\Phi \) be an Orlicz function of upper type \(p\in (0,\infty )\), \(0<q\le \infty \) and let b be a slowly varying function. For \(0<\kappa <\infty \), the functional \(\Vert \cdot \Vert _{L_{\Phi ,q,b}}\) can be discretized as follows
when \(0<q<\infty \), and
Proof
We only prove the case of \(0<q<\infty \):
Since the functions
are equivalent to non-increasing functions by Lemma 2 and Proposition 2, we get
which means
Conversely, according to Lemma 2, we have
Consequently, we conclude
Thus the estimation (2.2) is true. \(\square \)
Remark 4
Let \(\Phi \) be an Orlicz function of upper type \(p\in (0,\infty )\) and let b be a slowly varying function. Especially for \(\kappa =1\), it follows from the formula (2.2) that
If we take \(b\equiv 1\) for the above estimation, then we get
Moreover, if we consider the case \(\Phi (t)=t^p\) for \(0<p<\infty \), we obtain
Taking \(b\equiv 1\) and \(\Phi (t)=t^p\) for \(0<p< \infty \), we also obtain
Next we prove the following useful properties of Orlicz-Lorentz-Karamata spaces.
Proposition 3
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a slowly varying function. Then we have \(L_{p_+,q,b}\subseteq L_{\Phi ,q,b} \subseteq L_{p_-,q,b}\), i.e.,
Proof
Apparently, it follows from Lemma 4 and the definition of \(\Vert \cdot \Vert _{L_{\Phi ,q,b}}\) that (2.3) holds. \(\square \)
Proposition 4
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a slowly varying function. Then \(\Vert f\Vert _{L_{\Phi ,q,b}}\) and
are equivalent functionals.
Proof
For any measurable function f, there exists a sequence of non-negative simple functions \(\{f_n\}_{n \in \mathbb {N}}\) such that \(f_n \uparrow |f|\) a.e.. Moreover, \(d_{f_n} \uparrow d_f\) and \(f^*_n \uparrow f^*\). Therefore, by using Lebesgue monotone convergence theorem, it suffices to establish that the quasi-norm defined in (2.4) is equivalent to \(\Vert f\Vert _{L_{\Phi ,q,b}}\) for non-negative simple functions.
Now let
where \( \{A_i \}_{i=1}^N \) is a family of disjoint measurable sets and \( \{\alpha _j\}_{j=1}^N \subseteq \mathbb {R} \) satisfying \( 0 \le \alpha _j \le \alpha _i \) as \(1 \le i\le j \le N\). For any \(t\ge 0\), we have
where \(\alpha _{N+1}=0 \) and \( \beta _j=\sum \limits _{i=1}^j \mathbb {P}(A_i) \) for \(1\le j\le N\). Also, one can see that
where \(\beta _0=0\).
We first consider the case of \(q=\infty \). Since \(\Phi ^{-1}(t)\) is increasing on \((0,\infty )\) and \(\frac{\Phi ^{-1}(t)}{t^{1/p_+}}\) is equivalent to a non-decreasing function on \((0,\infty )\) (see Lemma 2), then we get
and
which implies
Now we consider the case of \(0<q<\infty \). Set
Then it follows from the Abel transformation that
Since \(\frac{\Phi ^{-1}(t)}{t^{1/p_+}}\) is equivalent to a non-decreasing function on \((0,\infty )\), and \(\frac{\Phi ^{-1}(t)}{t^{1/p_-}}\) is equivalent to a non-increasing function on \((0,\infty )\) (see Lemmas 1 and 2), then we have
Similarly, by Lemma 1,
Hence we get
Moreover, we have
Thus we get
which completes the proof. \(\square \)
Lemma 7
If \(\Phi \) is an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and b is a slowly varying function, then the Orlicz-Lorentz-Karamata spaces \(L_{\Phi ,q,b}\) are (quasi)-Banach spaces.
Proof
We only have to prove that
Indeed, it follows from Proposition 4 and \((f+g)^*(t_1+t_2)\le f^*(t_1)+g^*(t_2)\) for \(0\le t_1,t_2<\infty \) that
By applying Proposition 2, we have
According to \(\frac{1}{\Phi ^{-1}\big (1/(2t)\big )}\preceq \frac{1}{\Phi ^{-1}\big (1/t\big )}\) from Lemma 1 (iii), we obtain that
Thus we get that
This proves (2.5) as well as the lemma. \(\square \)
Let \((X,\Vert \cdot \Vert _X)\) be a (quasi)-normed space. A function \(f\in X\) is said to have absolutely continuous (quasi)-norm in X, if
for every sequence \((A_n)_{n\ge 0}\) satisfying \(\lim \limits _{n\rightarrow \infty }\mathbb {P}(A_n)=0\).
Proposition 5
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_-\le p_+<\infty \), \(0<q<\infty \) and let b be a slowly varying function. Then every \(f\in L_{\Phi ,q,b}\) has absolutely continuous quasi-norm.
Proof
Since \(f\in L_{\Phi ,q,b}\), for any \(\varepsilon >0\), there exists \(N_1\in \mathbb {N}\) such that
Let the sequence \((A_n)_{n\ge 0}\) satisfy \(\lim \limits _{n\rightarrow \infty }\mathbb {P}(A_n)=0\). There exists \(N_2\in \mathbb {N}\) such that for \(n\ge N_2\), \(\mathbb {P}(A_n)<\big (\frac{\varepsilon }{2^{N_1}\gamma _{b}(1)}\big )^{2p_+}\). Now let \(n\ge N_2\). By using Lemmas 6 and 4, we have
which yields that \(\Vert f\chi _{A_n}\Vert _{L_{\Phi ,q,b}} \preceq \varepsilon \). Hence, f has absolutely continuous quasi-norm. \(\square \)
Proposition 6
Let \( 0 < s \le q \le \infty \), \(\Phi \) be an Orlicz function of lower type \(p\in (s,\infty )\) and let b be a slowly varying function. Then we obtain that \(|||f |||_{L_{\Phi ,q,b}}\) and
are equivalent (quasi)-norms, where
Proof
Obviously, \(f^*(t)\le f_s^{**}(t)\) for \(t>0\). Thus it is sufficient to prove
From the definitions of \(\Vert \cdot \Vert _{L_{\Phi ,q,b,s}}\) and \(|||\cdot |||_{L_{\Phi ,q,b}}\), we see that
and
where
Applying Theorem 1.1 in [45], we know that, if the following estimation
is finite, then
where \(\alpha '=\alpha /(\alpha -1).\)
Hence, in order to get the inequality (2.6), we just compute the formula (2.7). Now let’s estimate the
respectively. The estimations are divided into the following two steps:
Step 1. We first consider the case of \(s<q\). That is
Since \(\frac{\Phi ^{-1}(t)}{t^{1/p}}\) is equivalent to non-increasing on \((0,\infty )\) via Lemma 1 (ii), we have
and
This means
Thus we see that (2.6) holds for \(s<q\).
Step 2. Now we consider the case \(s=q\). Then \(0<q<p\) and define
Similarly to Step 1, we obtain that
and
Thus we have
which implies that (2.6) holds for \(s=q\). This completes the proof. \(\square \)
Theorem 1
If \( 0 < q \le \infty \), \(0<s \le \min \{1,q\}\), \(\Phi \) is an Orlicz function of lower type \(p\in (s,\infty )\) and b is a slowly varying function, then \(|||\cdot |||_{L_{\Phi ,q,b}}\) is equivalent to an s-norm. Namely, there exists a finite C such that for any \(h_k\), \(k=1,2,\ldots ,n\),
Proof
It follows from p. 64 of [22] that for \( 0 < s \le 1 \),
According to Proposition 6, inequality (2.8) and Minkowski’s inequality, we have
Thus there exists a constant C independently on n such that
This completes the proof. \(\square \)
Combining this with Proposition 4 and Theorem 1, we get the following result.
Theorem 2
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<s<p_- \le p_+<\infty \), \(0<q\le \infty \), \(0<s \le \min \{1,q\}\) and let b be a slowly varying function. \(\Vert \cdot \Vert _{L_{\Phi ,q,b}}\) is equivalent to an s-norm.
2.3 Orlicz-Lorentz-Karamata Hardy martingale spaces
We recall some notations for martingales and define Orlicz-Lorentz-Karamata Hardy martingale spaces. Let \((\Omega ,\mathcal {F},\mathbb {P})\) be a probability space and \(\{\mathcal {F}_n\}_{n\ge 0}\) be a non-decreasing sequence of sub-\(\sigma \)-algebras of \(\mathcal {F}\) such that \(\mathcal {F}=\mathcal {\sigma }\big (\bigcup _{n\ge 0}\mathcal {F}_n\big )\). Recall that the expectation operator and the conditional expectation operator with respect to \({\mathcal {F}_n}\) are denoted by \(\mathbb {E}\) and \(\mathbb {E}_n\), i.e. \(\mathbb {E}_n f=\mathbb {E}(f|{\mathcal {F}_n})\) for any \(\mathcal {F}\)-measurable function f, respectively. A sequence of measurable functions \(f=(f_n)_{n\ge 0} \subseteq L_1\) is called a martingale with respect to \(\{\mathcal {F}_n\}_{n\ge 0}\) if \(\mathbb {E}_nf_{n+1}=f_n\) for every \(n\ge 0.\) Denote by \(\mathcal {M}\) the set of all martingales \(f=(f_n)_{n\ge 0}\) relative to \(\{\mathcal {F}_n\}_{n\ge 0}\) such that \(f_0=0\).
For \(f= (f_n)_{n\ge 0}\in \mathcal {M}\), denote its martingale difference by
We shall consider the following special martingale operators. For any martingale \(f\in \mathcal {M}\), we define the maximal functions, the square functions and the conditional square functions of f, respectively, as follows
Let \(\Lambda \) be the class of non-negative, non-decreasing and adapted sequences \(\lambda =(\lambda _n)_{n\ge 0}\) with respect to \(\{\mathcal {F}_n\}_{n\ge 0}\), i.e., \(\lambda _n\) is \(\mathcal {F}_n\) measurable for all \(n \in \mathbb {N}\). Define
Let \(\Phi \) be an Orlicz function, \(0<q\le \infty \) and let b be a slowly varying function. For \(f\in \mathcal {M}\), let
We define the Orlicz-Lorentz-Karamata Hardy martingale spaces via the martingale operators above in the following way:
Remark 5
If we consider the special case \(\Phi (t)=t^p\) for \(0<p<\infty \) with the notations above, then we obtain the definition of \(H_{p,q,b}^M\), \(H^S_{p,q,b}\), \(H^s_{p,q,b}\), \({\mathcal {Q}}_{p,q,b}\) and \({\mathcal {P}}_{p,q,b}\), respectively; see Ho [28]. If \(b\equiv 1\), we obtain the definition of \(H_{\Phi ,q}^M\), \(H^S_{\Phi ,q}\), \(H^s_{\Phi ,q}\), \({\mathcal {Q}}_{\Phi ,q}\) and \({\mathcal {P}}_{\Phi ,q}\), respectively; see Hao and Li [24]. In addition, if \(\Phi (t)=t^p\) for \(0<p<\infty \), then we obtain the definition of \(H_{p,q}^M\), \(H^S_{p,q}\), \(H^s_{p,q}\), \({\mathcal {Q}}_{p,q}\) and \({\mathcal {P}}_{p,q}\), respectively; see Jiao et al. [33, 37, 60]. Moreover, if \(p=q\), we obtain the martingale Hardy spaces \(H_q^M\), \(H^S_q\), \(H^s_q\), \({\mathcal {Q}}_q\) and \({\mathcal {P}}_q\), respectively; see Weisz [59, 60].
The bounded \(L_{\Phi ,q,b}\)-martingale spaces are defined by
In particular, for \(\Phi (t)=t^q\) and \(b\equiv 1\), we call these spaces bounded \(L_{q}\)-martingale spaces.
The stochastic basis \(\{\mathcal {F}_n\}_{n\ge 0}\) is said to be regular, if for \(n\ge 0\) and \(A\in \mathcal {F}_n\), there exists a \(B\in \mathcal {F}_{n-1} \) such that
where \(\mathcal {R}\) is a positive constant independent of n. Equivalently, \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular if there exists a constant \(\mathcal {R}>0\) such that
for all non-negative martingales \((f_n)_{n\ge 0}\). We refer to [42, 50, 60] for the theory of martingale Hardy spaces.
3 Doob’s maximal inequalities
Doob’s maximal inequality plays a central role in harmonic analysis, probability and ergodic theory. The classical Doob maximal inequality (see Doob [11], and also [19, 60]) states that
Doob used this inequality to prove the basic, almost sure convergence properties of martingales. Over the course of the past few decades, Doob’s maximal inequality has been rapidly developed to various function spaces. In this section, we study the Doob maximal inequalities on Orlicz-Lorentz-Karamata spaces.
Theorem 3
Let \( \Phi \) be an Orlicz function of lower type \(p\in (1,\infty )\), \(0<q\le \infty \) and let b be a slowly varying function. Then we have
As an application of Theorem 3, by using Proposition 4, we obtain the following result.
Theorem 4
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(1<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a slowly varying function. Then we have
In order to prove Theorem 3, we firstly present another characterization of the functional \(|||\cdot |||_{L_{\Phi ,q,b}}\) under suitable conditions.
Proposition 7
Let \(\Phi \) be an Orlicz function of lower type \(p\in (1,\infty )\), \( 0< q\le \infty \) and let b be a slowly varying function. Then \(|||f |||_{L_{\Phi ,q,b}}\) and
are equivalent (quasi)-norms. Here \(f^{**}\) denotes the maximal non-increasing rearrangement of f, which is defined as
To prove Proposition 7, we need the following characterization of Hardy-type inequalities.
Lemma 8
([3], Theorem 7) Let \(0<q\le 1\), v and w be non-negative measurable functions where v is assumed to be locally integrable on \((0,\infty )\). Then the weighted Hardy inequality
holds for all non-negative, non-increasing functions g on \((0,\infty )\) if and only if the following conditions hold
\((a)\quad \quad \quad \quad \quad \quad \quad \quad \sup \limits _{x>0}\frac{1}{V(x)^{1/q}}\Big (\int _{0}^{x}t^{q}w(t)dt\Big )^{1/q}<\infty \)
and
\((b)\quad \quad \quad \quad \quad \quad \quad \quad \sup \limits _{x>0}\frac{x}{V(x)^{1/q}}\Big (\int _{x}^\infty w(t)dt\Big )^{1/q}<\infty ,\)
where
Now we prove Proposition 7:
Proof
Obviously, \(f^*(t)\le f^{**}(t)\) for \(t>0\). Thus it is enough to prove that
Taking \(s=1\) in Proposition 6, it is easy to see that (3.1) holds when \(1\le q \le \infty \). Now we verify (3.1) for \(0<q<1\). From the definitions of \(\Vert \cdot \Vert _{L_{\Phi ,q,b,1}}\) and \(|||\cdot |||_{L_{\Phi ,q,b}}\), we see that
and
where
Clearly, g(x) is a non-negative and non-increasing function, and
Since \( \frac{\Phi ^{-1}(t)}{t^{1/{p}}}\) is equivalent to a non-increasing function on \((0,\infty )\) by Lemma 1, we have that \(\frac{(1/t)^{1/p}}{\Phi ^{-1}(1/t)}\) is also equivalent to a non-increasing function on \((0,\infty )\). Hence we obtain that, for \(x>0\),
and
where \(0<\varepsilon _0<1-\frac{1}{p}\). Thus we get the following estimation
Inequalities (3.2) and (3.3) and Lemma 8 imply
This completes the proof. \(\square \)
Lemma 9
([42], Theorem 3.6.3) Let \(f=(f_n)_{n\ge 0}\in L_1\). Then
Now we prove Theorem 3:
Proof
Obviously, \(|||f |||_{L_{\Phi ,q,b}}\le |||M(f) |||_{L_{\Phi ,q,b}}\). It follows from Proposition 7 and Lemma 9 that
This completes the proof of the theorem. \(\square \)
4 Atomic decomposition
Atomic decomposition is a powerful tool for dealing with interpolation theorems, duality theorems and some fundamental inequalities both in martingale theory and harmonic analysis. We refer to [5, 26, 34, 36, 41, 44, 51, 57, 60,61,62,63, 65,66,67] for more information on atomic decomposition. In this section we construct the atomic decomposition of Orlicz-Lorentz-Karamata Hardy martingale spaces. First, we give the definition of the atom.
Definition 4
Let \( \Phi \) be an Orlicz function and \(0<r\le \infty \). A measurable function, a, is called a \((1,\Phi ,r)\)-atom (resp. \((2,\Phi ,r)\)-atom, \((3,\Phi ,r)\)-atom) if there exists a stopping time \( \tau \in \mathcal {T} \) such that
-
1.
\(a_n=\mathbb {E}_n a =0\) if \(\tau \ge n\);
-
2.
\(\Vert s(a)\Vert _{L_r}\) (resp. \(\Vert S(a)\Vert _{L_r}\), \(\Vert M(a)\Vert _{L_r}\) ) \(\le \mathbb {P}(\tau<\infty )^{1/r}\Phi ^{-1}\big ( {1}/{\mathbb {P}(\tau <\infty )}\big ) \).
Notice that if we consider the special case \(\Phi (t)=t^p\) \((t\ge 0, \, 0<p<\infty )\), the \((i,\Phi ,\infty )\)-atom (\(i=1,2,3\)) is the same as Definition 2.1 in Weisz [60]. Denote by \(\mathcal {A}_i{(\Phi ,q,b,r)}\) the set of all sequences of triples \((a^k,\tau _k,\mu _k)_{k\in \mathbb {Z}}\) satisfying
(with the usual modification for \(q = \infty \)), where \((a^k)_{k\in \mathbb {Z}}\) are \((i,\Phi ,r)\)-atoms (\(i=1,2,3\)), \((\tau _k)_{k\in \mathbb {Z}}\subseteq \mathcal {T}\) are associated with \((a^k)_{k\in \mathbb {Z}}\), and there exists a positive constant \(\kappa \) such that
Definition 5
Let \( \Phi \) be an Orlicz function, \(0<q,r\le \infty \) and let b be a slowly varying function. The atomic martingale Orlicz-Lorentz-Karamata space \(H_{\Phi ,q,b}^{at,r,1}\) \(\big (\)resp. \(H_{\Phi ,q,b}^{at,r,2}\) or \(H_{\Phi ,q,b}^{at,r,3}\) \(\big )\) is defined to be the set of all \(f \in \mathcal {M} \) satisfying that there exists a sequence of triples, \((a^k,\tau _k,\mu _k)_{k\in \mathbb {Z}}\subseteq \mathcal {A}_1{(\Phi ,q,b,r)}\) \(\big (\)resp. \(\mathcal {A}_2{(\Phi ,q,b,r)}\) or \(\mathcal {A}_3{(\Phi ,q,b,r)}\) \(\big )\) such that for each \(n\ge 0\) and almost every \(\omega \in \Omega \),
Endow \(H_{\Phi ,q,b}^{at,r,1}\) \(\big (\)resp. \(H_{\Phi ,q,b}^{at,r,2}\) or \(H_{\Phi ,q,b}^{at,r,3}\) \(\big )\) with the (quasi)-norm
where the infimum is taken over all decompositions of f as above (with the usual modification for \(q = \infty \) \()\).
Remark 6
We claim that every \((1,\Phi ,\infty )\)-atom a is a \((1,\Phi ,r)\)-atom for \(0<r<\infty \). Indeed, from the definition of \((1,\Phi ,\infty )\)-atom, there is
This implies that
with respect to their (quasi)-norms respectively.
Theorem 5
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a slowly varying function. Then
with equivalent (quasi)-norms.
Proof
Let \(f \in H_{\Phi ,q,b}^s\). For any \(k\in \mathbb {Z}\), define
Apparently, \(\tau _k\) is a stopping time and \(\tau _k\le \tau _{k+1}\) for each \(k\in \mathbb {Z}\). It is easy to see that for each \(n\in \mathbb {N}\),
For every \(k\in \mathbb {Z}\) and \(n\in \mathbb {N}\), set
If \(\mu _k=0\) then let \(a_n^k=0\) for all \(k\in \mathbb {Z}\) and \(n\in \mathbb {N}\). Obviously, \((a_n^k)_{n \ge 0}\) is a martingale for each fixed \(k\in \mathbb {Z}\). Moreover, in view of the definition of \(\tau _k\), we have \(s(f^{\tau _k}) = s_{\tau _k}(f)\le 2^k\). Hence, by using the definition of \(\mu _k\), we obtain
That is, \((a_n^{k})_{n\ge 0}\) is an \(L_2\)-bounded martingale. Thus there exists an \(a^k\in L_2\) such that \(\mathbb {E}_na^k=a_n^k\) and
Furthermore, for \(\tau _k\ge n\),
Thus we conclude that \(a^k\) is really a \((1,\Phi ,\infty )\)-atom.
Suppose that \(0<q<\infty \). Since \(\{\tau _k<\infty \}=\{s(f)>2^k\}\), then it follows from Remark 4 that
Thus we deduce that
For the converse part, assume that the martingale f has the decomposition of (4.1). For an arbitrary integer \(k_0\), let
where
It follows from the sublinearity of the conditional square operator s that
Since \(a^k\) is a \((1,\Phi ,\infty )\)-atom for each \(k\in \mathbb {Z}\), we obtain
Combining this with formulas (2.2), (2.1) and (4.2), we can deduce that
On the other hand, since \(a^k\) is a \((1,\Phi ,\infty )\)-atom for each \(k\in \mathbb {Z}\), then we have \(s(a^k)=0\) on the set \(\{\tau _k=\infty \}\). This means
which implies
Moreover, applying with Theorem 2, there exists \(0<\alpha <\min \{1,p_-, q\}\) such that
Let \(0<\eta <1\). For \(1=\frac{\alpha }{q}+\frac{q-\alpha }{q}\), it follows from Hölder’s inequality and (4.5) that
Combining this with (4.3), we obtain that
This means
Taking the infimum over all the preceding decompositions of f of the form (4.1), we can conclude the result. The proof is similar for \(q=\infty \). This completes the proof of the theorem. \(\square \)
Remark 7
If \(\Phi \) is an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q< \infty \) and b is a slowly varying function, then the sum \( \sum ^N_{k=M} \mu _k a^k\) converges to f in \(H_{\Phi ,q,b}^s\) as \(M \rightarrow -\infty \), \(N\rightarrow \infty \). Indeed, since
then, by using (2.5), we get
It follows from \(s\big (f-f^{\tau _{N+1}}\big )^2=s(f)^2-s\big (f^{\tau _{N+1}})^2\) that \(s\big (f-f^{\tau _{N+1}}\big )\le s(f)\), \(s\big (f^{\tau _{M}}\big )\le s(f)\) and \(s\big (f-f^{\tau _{N+1}}\big )\longrightarrow 0\), \(s\big (f^{\tau _{M}}\big )\longrightarrow 0\) as \( M \longrightarrow -\infty \), \( N \longrightarrow \infty \). Then, by Proposition 5, we obtain
as \( M \longrightarrow -\infty \), \( N \longrightarrow \infty \). This means
Obviously, for \(k\in \mathbb {Z}\), \(a^k = (a^k_n)_{n\ge 0}\) (here \(a^k\) is a \((1, \Phi ,\infty )\)-atom) is \(L_2\)-bounded. Hence \( L_2 = H^s_2 \) is dense in \(H^s_{\Phi ,q,b}\).
Theorem 6
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a slowly varying function. If \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
with equivalent (quasi)-norms.
Proof
Let \(f\in H_{\Phi ,q,b}^M\). For each fixed \(k\in \mathbb {Z}\), the stopping times \(v_k\) are defined by
Define
Since \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, we can see that
Define a new family of stopping times by
Then \(v_k(\omega )=n\) means \(\omega \in F_n^k\), which deduces \(\tau _k(\omega )\le n-1\). In other words, \(\tau _k< v_k\) on the set \(\{v_k\ne \infty \}\). Clearly, the sequence of these stopping times \((\tau _k)_{k\in \mathbb {Z}}\) is non-decreasing. Moreover, by applying Proposition 3, we get that
where the first inequality was proved in Lemma 4 of [57]. Then
which means that
Hence \(\lim \limits _{k\rightarrow \infty }\tau _k=\infty \) a.e. and for \(n\in \mathbb {N}\), \(\lim \limits _{k\rightarrow \infty }f_n^{\tau _k}=f_n\) a.e. Let \((a^k)_{k\in \mathbb {Z}}\) and \((\mu _k)_{k\in \mathbb {Z}}\) be defined as in the proof of the Theorem 5, then the conclusions \(f_n=\sum _{k\in \mathbb {Z}}\mu _k \mathbb {E}_{n}a^k\) and
still hold.
We omit the proof of the converse part, since it is similar to the proof of Theorem 5. The proof is completed. \(\square \)
Similarly to the method of the proof above, it is easy to get the atomic decomposition of \(H^S_{\Phi ,q ,b}\) as follows.
Theorem 7
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a slowly varying function. If \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
with equivalent (quasi)-norms.
Next, we consider the atomic decompositions of \( \mathcal {Q}_{\Phi ,q,b}\) and \(\mathcal {P}_{\Phi ,q,b}\).
Theorem 8
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a slowly varying function. Then
with equivalent (quasi)-norms.
Proof
The proof is similar to that of Theorem 5, so we only sketch it. If \(f=(f_n)_{n \ge 0} \in \mathcal {Q}_{\Phi ,q,b}\) (resp. \(\mathcal {P}_{\Phi ,q,b}\)), then the stopping times \(\tau _k\) are defined by
where \((\lambda _n)_{n\ge 0}\) is an adapted, non-decreasing sequence such that almost everywhere \(|S_n(f)| \le \lambda _{n-1}\) (resp. \(|f_n| \le \lambda _{n-1}\)) and \(\lambda _{\infty }\in L_{\Phi ,q,b}.\) Let \((a^k)_{k\in \mathbb {Z}}\) and \((\mu _k)_{k\in \mathbb {Z}}\) be defined as in the proof of Theorem 5. Then the conclusions
and
still hold.
To prove the converse part, let
Then \((\lambda _n)_{n\ge 0}\) is a non-negative, non-decreasing and adapted sequence. It follows from the first condition of the atom that
For any given integer \(k_0\), set
where
and
By replacing \(s(T_1)\) and \(s(T_2)\) in the proof of Theorem 5 with \(\lambda _\infty ^{\top }\) and \(\lambda _\infty ^{\bot }\), respectively, we get that \(f\in \mathcal {Q}_{\Phi ,q,b}\) (resp. \(f\in \mathcal {P}_{\Phi ,q,b}\)) and
where the infimum is taken over all decompositions of f of the form (4.1). This completes the proof of the theorem. \(\square \)
In particular, if we consider the case \( \Phi (t)=t^p \), then the following result holds:
Corollary 1
Let \(0<p<\infty \), \(0<q \le \infty \) and b be a slowly varying function. We get the atomic decomposition of Lorentz-Karamata Hardy martingale spaces \(H_{p,q,b}^s\), \(\mathcal {Q}_{p,q,b}\) and \(\mathcal {P}_{p,q,b}\), respectively.
Remark 8
We refer the reader to [38] for the atomic decomposition of Lorentz-Karamata Hardy martingale spaces in the case of \(0<p<\infty \), \(0<q\le \infty \) and b is a non-decreasing slowly varying function. It is noteworthy that the slowly varying function b does not need to satisfy the non-decreasing condition in Corollary 1. Hence our results improve the atomic decomposition of Lorentz-Karamata Hardy martingale spaces in [38].
Especially for \(b\equiv 1\), we obtain the following conclusion:
Corollary 2
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \) and \(0<q\le \infty \). Then we obtain the atomic decomposition of Orlicz-Lorentz Hardy martingale spaces \(H_{\Phi ,q}^s\), \(\mathcal {Q}_{\Phi ,q}\) and \(\mathcal {P}_{\Phi ,q}\), respectively.
Remark 9
We refer the reader to [24] for the atomic decomposition of Orlicz-Lorentz Hardy martingale spaces in the case of \(0<q\le 1\) and \(\Phi \) is an Orlicz function with \(q_{\Phi ^{-1}} <1/q\). Notice that, there is no need for the condition \(q_{\Phi ^{-1}} <1/q\) in Corollary 2. Hence we also extend and improve the atomic decomposition of Orlicz-Lorentz Hardy martingale spaces in [24].
5 Martingale inequalities
In this section, we show some martingale inequalities between different Orlicz-Lorentz-Karamata Hardy martingale spaces. The method we used here is to establish a sufficient condition for a \(\sigma \)-sublinear operator to be bounded from martingale Orlicz-Lorentz-Karamata Hardy spaces to Orlicz-Lorentz-Karamata spaces.
Let us recall that an operator \(T:X\rightarrow Y\) is said to be \(\sigma \)-sublinear if for any constant c,
where X is a martingale space and Y is a measurable function space.
Lemma 10
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a non-decreasing slowly varying function. Suppose that \(\max \{p_+,1\}<r<\infty \). If \(T:H_r^s\rightarrow L_r\) is a bounded \(\sigma \)-sublinear operator and
for every \((1,\Phi ,\infty )\)-atom a associated with the stopping time \(\nu \), then for \(f\in H_{\Phi ,q,b}^s\),
Remark 10
Let \(0<L<\infty \) and \(\widetilde{\Phi }(t)=\Phi \big (t^{L}\big )\) for \(t\in [0,\infty )\). If \(\Phi \) is an Orlicz function of lower type \(p_-\) and upper type \(p_+\), then \(\widetilde{\Phi }\) is an Orlicz function of lower type \(Lp_-\) and upper type \(Lp_+\). Under the conditions of Lemma 10, we have
where \(0<L<r/p_+\). Indeed, it was proved in [66] that T is bounded from \(H_{\widetilde{\Phi }}^s\) to \(L_{\widetilde{\Phi }}\). Hence we obtain that
Moreover, since a is a \((1,\Phi ,\infty )\)-atom, we get
Now we turn to prove Lemma 10:
Proof
Let \(f\in H_{\Phi ,q,b}^s\). According to Theorem 5, there exist a sequence \((a^k)_{k\in \mathbb {Z}}\) of \((1,\Phi ,\infty )\)-atoms associated with stopping times \((\tau _k)_{k\in \mathbb {Z}}\) and a sequence of real numbers
such that
and
For an arbitrary integer \(k_0\), set
Then by the \(\sigma \)-sublinearity of T, there is
Let \(0<\varepsilon <\min \{\varrho ,p_-,q\}\), \(1<L<\min \{\frac{r}{p_+},\frac{1}{\varepsilon }\}\) and \(0<\sigma <1-\frac{1}{L}\), where \(\varrho \) is the same constant as in Remark 2. By using Hölder’s inequality, we conclude
where \(\frac{1}{L'}+\frac{1}{L}=1\). According to Remark 2 and Remark 10, we have
Hence, we can choose \(\delta >0\) such that \(1<\delta <L(1-\sigma )\). By Hölder’s inequality, we have
Let \(b_1(t)=b\big (\Phi (t)\big )\) for \(t\in [1,\infty )\). From Proposition 1, \(b_1\) is a slowly varying function. Clearly, \(b_1\) is non-decreasing. Hence we get
Set \(0<z<\frac{\delta -1}{\delta }\). Since \(t^z\gamma _{b_1}(t)\) is equivalent to a non-decreasing function, then
where \(m<k_0\). Combining this with Remark 4 and the Abel transformation, we have
and further
Now, we estimate \(D_2\). Since \(\{|T(a^k)|>0\}\subseteq \{\tau _k<\infty \}\) for each \(k\in \mathbb {Z}\), we have
Moreover, by using Theorem 2, there exists \(0<s<\min \{1,p_-, q\}\) such that
Let \(0<\eta <1\). For \(1=\frac{s}{q}+\frac{q-s}{q}\), it follows from Hölder’s inequality and (5.3) that
It follows from Remark 4 and Abel’s transformation that
Thus, combining with (5.1) and (5.4), we have
The proof is completed now. \(\square \)
Similar to the method of the proof above, we can present the following results.
Lemma 11
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a non-decreasing slowly varying function. Suppose that \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular and \(\max \{p_+,1\}<r<\infty \). If \(T:H_r^S\rightarrow L_r \; (\text {resp.}\ H_r^M\rightarrow L_r)\) is a bounded \(\sigma \)-sublinear operator and
for every \((2,\Phi ,\infty )\)-atom (resp. \((3,\Phi ,\infty )\)-atom) a associated with the stopping time \(\nu \), then for \(f\in H_{\Phi ,q,b}^S\) \((\text {resp.}\; f\in H_{\Phi ,q,b}^M)\),
Lemma 12
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a non-decreasing slowly varying function. Suppose that \(\max \{p_+,1\}<r<\infty \). If \(T:H_r^S\rightarrow L_r\;(\text {resp.}\ H_r^M\rightarrow L_r)\) is a bounded \(\sigma \)-sublinear operator and
for every \((2,\Phi ,\infty )\)-atom (resp. \((3,\Phi ,\infty )\)-atom) a associated with the stopping time \(\nu \), then for \(f\in \mathcal {Q}_{\Phi ,q,b}\) \((\text {resp.}\ f\in \mathcal {P}_{\Phi ,q,b})\),
To get the martingale inequalities for martingale Orlicz-Lorentz-Karamata Hardy spaces, we need the following lemma.
Lemma 13
([19, 42, 60]) Let f be a martingale. Then
Moreover, if the stochastic basis \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
Theorem 9
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a non-decreasing slowly varying function. Then
Moreover, if \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
with equivalent norms.
Proof
It is clear that the operators s, S, and M are all \(\sigma \)-sublinear and
and
where \(a_i\;(i=1,2,3)\) is and \((i,\Phi ,\infty )\)-atom associated with the stopping time \(\nu _i\), respectively.
Lemma 13 gives that the operators \(M:H_2^s\rightarrow L_2\) and \(S:H_2^s\rightarrow L_2\) are bounded. Hence, by using Lemma 10, we get that the inequalities in (5.6) hold if \(0<p_-\le p_+<2\).
The inequalities in (5.7) can be directly get from the definitions of \(\mathcal {P}_{\Phi ,q,b}\) and \(\mathcal {Q}_{\Phi ,q,b}\).
According to Doob’s maximal inequality and Brukholder-Davis-Gundy’s inequality (see [60], Theorem 2.12), namely,
we obtain the operators \(M:H_r^S\rightarrow L_r\) and \(S:H_r^M\rightarrow L_r\) are bounded. Hence according to Lemma 12, the inequalities in (5.8) hold.
Let \(\max \{p_+,2\}<r<\infty \). It follows from Lemma 13 that \(s:H_r^M\rightarrow L_r\) and \(s:H_r^S\rightarrow L_r\) are bounded. Then by Lemma 12, we obtain the inequalities in (5.9).
To prove (5.10), we take \(f=(f_n)_{n\ge 0}\in \mathcal {Q}_{\Phi ,q,b}\). This means there exists \((\lambda _n^{(1)})_{n\ge 0}\in \Lambda [\mathcal {Q}_{\Phi ,q,b}](f)\) such that \(S_n(f)\le \lambda _{n-1}^{(1)}\) with \(\lambda _\infty ^{(1)}\in L_{\Phi ,q,b}\). Since
and \((\lambda _{n}^{(1)}+M_{n}(f))_{n\ge 0}\in \Lambda [\mathcal {Q}_{\Phi ,q,b}](f)\), by using the second inequality of (5.8), we get
If \(f=(f_n)_{n\ge 0}\in \mathcal {P}_{\Phi ,q,b}\), then there exists \((\lambda _n^{(2)})_{n\ge 0}\in \Lambda [\mathcal {P}_{\Phi ,q,b}](f)\) such that \(|f_n|\le \lambda _{n-1}^{(2)}\) with \(\lambda _\infty ^{(2)}\in L_{\Phi ,q,b}\). Since
and \((\lambda _{n}^{(2)}+S_{n}(f))_{n\ge 0}\in \Lambda [\mathcal {P}_{\Phi ,q,b}](f)\), by using the first inequality of (5.8), we have
Hence, inequality (5.10) holds and it yields that \(\mathcal {P}_{\Phi ,q,b}=\mathcal {Q}_{\Phi ,q,b}\).
Let \(\{\mathcal {F}_n\}_{n\ge 0}\) be regular and \(\max \{p_+,1\}<r<\infty \). Combining (5.5) with Lemma 10 and Lemma 11, we have
It follows from Theorem 6 and Theorem 8 that \(H_{\Phi ,q,b}^M=\mathcal {P}_{\Phi ,q,b}\). This and (5.10) imply
The proof of this theorem is completed. \(\square \)
The following corollary follows from Theorems 4 and 9.
Corollary 3
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(1<p_- \le p_+<\infty \), \(0<q\le \infty \) and let b be a non-decreasing slowly varying function. Then \(H_{\Phi ,q,b}^M\) is equivalent \(L_{\Phi ,q,b}\). If \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then the spaces \(H_{\Phi ,q,b}^S\), \(\mathcal {Q}_{\Phi ,q,b}\), \(\mathcal {P}_{\Phi ,q,b}\), \(H_{\Phi ,q,b}^M\), \(H_{\Phi ,q,b}^s\) and \(L_{\Phi ,q,b}\) are all equivalent.
Taking \(b\equiv 1\) in Theorem 9, we obtain the fundamental martingale inequalities on Orlicz-Lorentz Hardy spaces.
Corollary 4
Let \(\Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<\infty \) and \(0<q\le \infty \). Then
Moreover, if \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then the spaces \(H_{\Phi ,q}^S\), \(\mathcal {Q}_{\Phi ,q}\), \(\mathcal {P}_{\Phi ,q}\), \(H_{\Phi ,q}^M\) and \(H_{\Phi ,q}^s\) are all equivalent. If \(1<p_- \le p_+<\infty \), then these spaces are also equivalent to \(L_{\Phi ,q}\).
Note that this result was proved for \(\Phi (t)=t^p\), i.e., for Hardy and Hardy-Lorentz spaces, in [59, 60].
6 Duality theorems
The goal of this section is to establish the dual spaces of Orlicz-Lorentz-Karamata Hardy martingale spaces \(H^s_{\Phi ,q,b}\). The dual spaces of \(H_p^s\) and \(H_{p,q}^s\) are known. More exactly, the dual spaces of \(H_p^s\) (\(0<p\le 1\)) are the BMO spaces when \(p=1\) and the Lipschitz spaces \(\Lambda _2(1/p-1)\) where \(0<p<1\). Moreover, the dual spaces of Lorentz Hardy martingale spaces \(H_{p,q}^s\) (\(0<p\le 1\), \(1<q<\infty \)) are the generalized BMO martingale spaces \(BMO_{2,q}(1/p-1)\) (see [37, 59, 60]). In this section, we generalize these results for the \(H^s_{\Phi ,q,b}\) spaces when \(\Phi \) is an Orlicz function and \(0<q<\infty \).
To state the duality, we need some more notations in this section. Let \(1\le p < \infty \), and let \(L_p^0\) be the set of all \(f \in L_p\) such that \(\mathbb {E}_0f = 0\). For any \(f \in L_p^0\), let \(f_n = \mathbb {E}_n f\). Then, it is known that \((f_n )_{n\ge 0}\) is an \(L_p\)-bounded martingale in \(\mathcal {M}\) and converges to f in \(L_p\) (see [49]). Now we define the generalized BMO martingale spaces as follows.
Definition 6
Let \(\Phi \) be an Orlicz function, \(1\le p < \infty \) and let b be a slowly varying function. The generalized BMO martingale space \(BMO_{p,\Phi ,b}\) is defined by
where
Note that if \(\Phi (t) = t^{\frac{1}{\lambda +1}}\), \(\lambda \in (-1,\infty )\), we have the martingale Campanato spaces \(\mathcal {L}_{p,\lambda }\) introduced in [46]. Moreover, if \(\Phi (t)=t^p\) for \(0<p \le 1\), the spaces \(\mathcal {L}_{2,\Phi }\) coincide with \(BMO_{2,\alpha }\), where \(\alpha =1/p-1\). See [37, 59, 60] for the notation \(BMO_{2,\alpha }\). In particular, take \(\Phi (t)=t\), the \(\mathcal {L}_{2,\Phi }\) can be reduced to the Banach space BMO (Bounded Mean Oscillation, see [27, 59, 60]). It is well known that the dual space of \( {H^s_1}\) is BMO in analogy with the result of Fefferman [16] for the classical case. We refer to [19] and [42, 59, 60] for the facts above.
Now we can identity the dual space of \(H^s_{\Phi ,q,b}\) (\(0<q\le 1\)):
Theorem 10
If \(\Phi \) is an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<2\), \(0<q\le 1\) and b is a slowly varying function, then the dual space of \(H_{\Phi ,q,b}^s\) is \(BMO_{2,\Phi ,b}\). That is
To prove the theorem above, we need a classical result from [12], Theorem 3.4.48] as follows.
Lemma 14
Let \(p_1,p_2,q_1,q_2 \in (0,\infty )\) with \(p_2 < p_1\) and let \(b_1,b_2\) be slowly varying functions. Then
with respect to (quasi)-norms.
Now we turn to prove Theorem 10:
Proof
Since \( \Phi \) is an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<2\), by using Lemma 14 and Proposition 3, we obtain that
In addition, Remark 7 assures that \(L_2\) is dense in \(H_{\Phi ,q,b}^s\). We shall prove that
is a continuous linear functional on \(H_{\Phi ,q,b}^s\), where \(g\in BMO_{2,\Phi ,b}\subseteq L_2\) is arbitrary. We take the same stopping time \(\tau _k\), atoms \(a^k\) and real numbers \(\mu _k\) (\(k\in \mathbb {Z}\)) as we did in Theorem 5. Modifying slightly the proof of that theorem, we obtain that
holds and also in \(L_2\) norm if \(f\in L_2\). Since \(g\in L_2\), we have
By the definition of atom \(a^k\), we get that
Hence, applying Hölder’s inequality, we conclude that
Since \( 0 < q \le 1\), it follows from concavity that
Therefore, Theorem 5 gives that
Conversely, let \(\ell \) be any continuous functional on \(H^s_{\Phi ,q,b}\). We show that there exists \(g \in BMO_{2,\Phi ,b}\) such that
First we note that, by
the space \(L_2\) can be embedded continuously in \(H^s_{\Phi ,q,b}\). Consequently, there exists a \(\widetilde{g}\in L_2\) such that
Let \(g =\widetilde{g}-\mathbb {E}_0[\widetilde{g}]\). Then \(g\in L_2^0\) and \(\mathbb {E}[f \widetilde{g}] = \mathbb {E}[fg]\) for \(f\in L_2^0\), that is,
Next we show that \(g \in BMO_{2,\Phi ,b}\) and that \(\Vert g\Vert _{BMO_{2,\Phi ,b}}\le C \Vert \ell \Vert \). Indeed, for any \(\tau \in \mathcal {T}\) with \(\mathbb {P}(\tau <\infty ) \ne 0\), suppose that
The function \(\varphi \) is not necessarily a \((1,\Phi ,\infty )\)-atom, however, it satisfies the condition (1) of Definition 4, namely,
Let \(h=g-g^\tau \). Note that \(s(h)=s(h)\chi _{\{\tau < \infty \}}\) also holds. Hölder’s inequality assures that
Since \(\frac{\Phi ^{-1}(t)}{t^{1/p_+}}\) is equivalent to a non-decreasing function on \((0,\infty )\), we have
Thus there exists a positive constant c such that
which means
Hence, we get
Thus, we obtain
Therefore, \(g \in BMO_{2,\Phi ,b}\). Taking the supremum over all stopping times, we proved (6.3). Hence, we establish the embedding \(\big (H^s_{\Phi ,q}\big )^* \subseteq BMO_{2,\Phi ,b}\) with respect to (quasi)-norms. This completes the proof of the theorem. \(\square \)
As a consequence, Theorem 1.2 in [38] is also true without the assumption that the slowly varying function b is non-decreasing. More precisely,
Corollary 5
Let \(0<p<2\), \(0<q\le 1\) and let b be a slowly varying function. Then the dual space of \(H^s_{p,q,b}\) is \(BMO_{2,b}(\alpha )\) with \(\alpha =1/p-1\).
Taking \(\Phi (t)=t^p\) for \(t\in [0,\infty )\) in Theorem 10, we can see that Corollary 5 holds. If we consider the special case \(b\equiv 1\) in Theorem 10, we conclude the following result.
Corollary 6
Let \( \Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<2\) and \(0<q\le 1\). Then the dual space of \(H_{\Phi ,q}^s\) is \(BMO_{2,\Phi }\). That is
There is no restricted condition \(q_{\Phi ^{-1}}<1/q\) in Corollary 6. Hence, it generalizes the very recent result in [24]. For another application of Theorem 10, let \(\Phi (t)=t^p\) and \(b\equiv 1\). Then the dual space of Lorentz Hardy martingale space is \(BMO_2(\alpha )\), where \(\alpha =1/p-1\).
Corollary 7
([37]) If \( 0< p <2\) and \( 0 <q\le 1\), then the dual space of \(H_{p,q}^s\) is \(BMO_2(\alpha )\), where \(\alpha =1/p-1\).
Note that the conclusion of Corollary 7 requires not the restriction \(q<p\). So this generalizes the result in [28].
Now, we consider the dual of Orlicz-Lorentz-Karamata Hardy martingale spaces \(H^s_{\Phi ,q,b}\) for \(1<q<\infty \).
Definition 7
Let \(\Phi \) be an Orlicz function, \(1\le p < \infty \), \(0<q<\infty \) and let b be a slowly varying function. The generalized BMO martingale space \(BMO_{p,q,\Phi ,b}\) is defined by
where
and the supremum is taken over all stopping time sequences \(\{\nu _k\}_{k\in \mathbb {Z}}\) such that \(\big \{ 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big \}_{k\in \mathbb {Z}}\in l_q\).
It follows from the idea of Proposition 4.6 in [35] and Proposition 6.7 in [63] that \(BMO_{p,q,\Phi ,b}\) and \(BMO_{p,\Phi ,b}\) have the following connection.
Proposition 8
Let \(\Phi \) be an Orlicz function, \(1\le p < \infty \), \(0<q<\infty \) and let b be a slowly varying function. Then
Moreover, if \(0<q \le 1\), then \(\Vert f\Vert _{BMO_{p,\Phi ,b}} = \Vert f\Vert _{BMO_{p,q,\Phi ,b}}\).
Proof
Let \(f\in BMO_{p,q,\Phi ,b}\). For any given \(\tau \in \mathcal {T}\) and fixing \(n_0\in \mathbb {Z}\), we set, for \(k\in \mathbb {Z}\),
It is easy to see that
This implies \(\Vert f\Vert _{BMO_{p,\Phi ,b}}\le \Vert f\Vert _{BMO_{p,q,\Phi ,b}}\).
For the converse, let \(0<q\le 1\), \(f\in BMO_{p,\Phi ,b}\) and
Then we get that
Taking over all stopping time sequences \(\{\nu _k\}_{k\in \mathbb {Z}}\) associated with
we have
Thus, if \(0<q \le 1\), then \(\Vert f\Vert _{BMO_{p,\Phi ,b}} = \Vert f\Vert _{BMO_{p,q,\Phi ,b}}\). \(\square \)
Remark 11
In Definition 7, if \(\Phi (t)=t^{1/(1+\alpha )}\) for \(\alpha \ge 0\), then the generalized BMO martingale space \(BMO_{p,q,\Phi ,b}\) goes back to \(BMO_{p,q,b}(\alpha )\), which was introduced in [38]. In addition, if \(b\equiv 1\), \(BMO_{p,q,b}(\alpha )\) is \(BMO_{p,q}(\alpha )\), which was introduced in [37].
Now we prove that the dual of \(H_{\Phi ,q,b}^s\) is \(BMO_{p,q,\Phi ,b}\) if \(1<q<\infty \).
Theorem 11
If \( \Phi \) is an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+<2\), \( 1<q<\infty \) and b is a slowly varying function, then the dual space of \(H_{\Phi ,q,b}^s\) is \(BMO_{2,q,\Phi ,b}\). That is
Proof
Let \(g\in BMO_{2,q,\Phi ,b}\subseteq L_2\) and define the linear functional
For any \(f \in L_2\), Theorem 5 provides a sequence \((a^k )_{k\in \mathbb {Z}}\) of \((1, \Phi ,\infty )\)-atoms and a sequence of real numbers \((\mu _k )_{k\in \mathbb {Z}}\) satisfying
where \(\kappa \) is a positive constant and \((\tau _k)_{k\in \mathbb {Z}}\) is the corresponding stopping time sequence. Moreover, \(f =\sum _{k\in \mathbb {Z}} \mu _ka^k\) and
Similarly to (6.2), we get
Since \(L_2\) is dense in \(H^s_{\Phi ,q,b}\) from Remark 7, the functional \(\ell \) can be uniquely extended to a continuous linear functional on \(H^s_{\Phi ,q,b}\).
Conversely, let \(\ell \in ( H_{\Phi ,q,b}^s)^*\). According to Theorem 5 and Remark 6, we have
This means that
Hence there exists a \(g \in L_2\) such that
and \(\ell \) can be extended to a continuous functional \(\tilde{\ell }\) on \(H_{\Phi ,q,b}^{at,2,1}\) such that \(\Vert \ell \Vert =\Vert \tilde{\ell }\Vert \). Let \(\{\nu _k\}_{k\in \mathbb {Z}}\) be an arbitrary stopping time sequence such that
Now we set
We claim that \(h_k\) is a \((1,\Phi ,2)\)-atom for each \(k\in \mathbb {Z}\). Indeed, if \(n\le \nu _k\), then clearly \(\mathbb {E}_nh_k=0\). Moreover,
It follows from formulae (6.4) and (6.5) that
and
Let
Consequently,
Thus we obtain
Taking \(N \longrightarrow \infty \) and the supremum over all stopping time sequences such that
we conclude
The proof is complete. \(\square \)
Theorem 11 improves the recent results [37, 38]. To be more specific, we obtain the conclusion of Theorem 1.5 in [38] without the condition that b is non-decreasing. That is, if we consider the case \(\Phi (t)=t^p\) for \(t\in [0,\infty )\) in Theorem 10, one get the following result:
Corollary 8
Let \(0<p<2\), \(1<q<\infty \) and let b be a slowly varying function. Then the dual space of \(H^s_{p,q,b}\) is \(BMO_{2,q,b}(\alpha )\) with \(\alpha =1/p-1\).
In particular, if \(\Phi (t)=t^p\) for \(t\in [0,\infty )\) and \(b\equiv 1\) in Theorem 11, we obtain the next corollary, which was proved in [37, 59, 60].
Corollary 9
If \(0<p<2\) and \(1<q<\infty \), then the dual space of \(H_{p,q}^s\) is \(BMO_{2,q}(\alpha )\), where \(\alpha =1/p-1\).
7 The generalized John-Nirenberg theorem
In this section, we prove the generalized John-Nirenberg theorem when the stochastic basis \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular.
Theorem 12
Let \( \Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+\le 1\), \(1< p,\,q < \infty \) and let b be a slowly varying function. If the stochastic basis \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
Proof
Let \(p'\) be the conjugate number of p, that is, \(1/p'+1/p=1\). Obviously, \(p'>1\). Firstly we claim that \(L_{p'}\subseteq H^s_{\Phi ,q,b}\). Indeed, for any \(f\in L_{p'}\), it follows from Proposition 3, Lemma 14 and Corollary 4 that
On the basis of Theorem 5, there exist a sequence \((a^k )_{k\in \mathbb {Z}}\) of \((1, \Phi ,\infty )\)-atoms and a sequence of real numbers \((\mu _k )_{k\in \mathbb {Z}}\) satisfying
where \(\kappa \) is a positive constant and \((\tau _k)_{k\in \mathbb {Z}}\) is the corresponding stopping time sequence, such that
and
Similarly to the proof of Theorem 10, we conclude that (7.1) holds also in the \(L_{p'}\)-norm. Then for any given \(g\in BMO_{p,q,\Phi ,b}\subseteq L_p\), we have
According to the Hölder inequality, we get
Thus \(\ell _g\) can be extended to a continuous functional on \(H^s_{\Phi ,q,b}\).
To prove the converse, let \(\ell \in ( H_{\Phi ,q,b}^s)^*\). It follows from Theorem 5, Remark 6 and Corollary 4 that
This means that
Hence there exists a \(g \in L_p\) such that
and \(\ell \) can be extended to a continuous functional \(\tilde{\ell }\) on \(H_{\Phi ,q,b}^{at, p', 1}\) such that \(\Vert \ell \Vert =\Vert \tilde{\ell }\Vert \). Let \(\{\nu _k\}_{k\in \mathbb {Z}}\) be an arbitrary stopping time sequence such that
Based on the sequence \(\{\nu _k\}_{k\in \mathbb {Z}}\), we set
Now we claim that \({h_k}/{c_0}\) is a \((1,\Phi ,{p'})\)-atom for each \(k\in \mathbb {Z}\), where \(c_0>1\) is the constant only dependents on p. Indeed, via Corollary 4, if \(n\le \nu _k\), then \(\mathbb {E}_nh_k=0\) and
It follows from formulas (7.2) and (7.3) that
and
For an arbitrary positive integer N, set
Consequently,
Thus we obtain
Taking \(N \longrightarrow \infty \) and the supremum over all stopping time sequences such that \(\{ 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\}_{k\in \mathbb {Z}}\in l_q\), we conclude \(\Vert g\Vert _{BMO_{p,q,\Phi ,b}}\le \Vert \ell \Vert \). The proof is complete. \(\square \)
Obviously, combining Theorem 12 with Theorem 11, we obtain the conclusion as follows.
Theorem 13
Let \( \Phi \) be an Orlicz function of lower type \(p_-\) and upper type \(p_+\) satisfying \(0<p_- \le p_+\le 1\), \(1<p,\,q<\infty \) and let b be a slowly varying function. If the stochastic basis \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
As an application of Theorem 13, combining with Remark 11, we get the following result:
Corollary 10
Let \(\alpha \ge 0\), \(1<p,\,q<\infty \) and b be a slowly varying function. If the stochastic basis \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
Remark 12
Note that, in the corollary above, the slowly varying function b is not necessarily non-decreasing as in the result of [38].
Remark 13
Theorem 13 and Corollary 10 are similar to the vector-valued characterizations of BMO on Euclidean space in [29].
For another application of Theorem 13, let \(\Phi (t)=t^p\) and \(b\equiv 1\). It is clear that we get the generalized John-Nirenberg theorem for Lorentz Hardy martingale spaces \(H^s_{p,q}\), which was proved in [37, 59, 60].
Corollary 11
Let \(\alpha \ge 0\), \(1<p,\,q<\infty \). If the stochastic basis \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular, then
If \(b\equiv 1\) and the stopping time sequence \(\{\nu _k\}_{k\in \mathbb {Z}}\) is a sequence whose one element is a stopping time \(\nu \) and the others are \(\infty \) in Definition 7, then Theorem 13 returns to the famous result, the John-Nirenberg characterization of BMO as
8 Boundedness of generalized fractional integral operators
We now extend the boundedness of generalized fractional integrals on Orlicz-Lorentz-Karamata Hardy martingale spaces. In this section, we always assume that \((\Omega ,\mathcal {F},\mathbb {P})\) is a complete and non-atomic probability space. Here, we suppose that every \(\sigma \)-algebra \(\mathcal {F}_n\) is generated by countable many atoms. \(B\in \mathcal {F}_n\) is called an atom if for any \(A\subseteq B\) with \(A\in \mathcal {F}_n\) satisfying \(\mathbb {P}(A)<\mathbb {P}(B)\), we have \(\mathbb {P}(A)=0\). Denote by \(A(\mathcal {F}_n)\) the set of all atoms in \(\mathcal {F}_n\). Without loss of generality, we always suppose that the constant in (2.9) satisfies \(\mathcal {R}\ge 2\). Now we give the definition of generalized fractional integrals as follows.
Definition 8
Let \(\phi \) be an Orlicz function. For \(f=(f_n)_{n\ge 0}\in \mathcal {M}\), the generalized fractional integral \(I_\phi f=\big ((I_\phi f)_n\big )_{n\ge 0}\) of f is defined by
where
Moreover, if \( \phi (t)=t^\alpha \) (\(\alpha >0\)), then we say that \( I_\phi f\) =\( I_\alpha f\) is the fractional integral (of order \(\alpha \)) of martingale f which was introduced in [23, 37, 54].
Remark 14
Note that \(I_\phi f=\big ((I_\phi f)_n\big )_{n\ge 0}\) is also a martingale, more exactly, a martingale transform introduced by Burkholder. Especially, if \(\Omega = [0, 1]\) and the \(\sigma \)-algebra \(\mathcal {F}_n\) is generated by the dyadic intervals of [0, 1], then \(I_\alpha f\) is closely related to a class of multiplier transformations of Walsh-Fourier series [58]. We refer to [55] for the classical fractional integrals.
In order to prove the boundedness of generalized fractional integrals on Orlicz-Lorentz-Karamata Hardy martingale spaces, we need the following lemmas.
Lemma 15
Let \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular and \(\phi \) be an Orlicz function of upper type \(p\in (0,1)\). If \(f=(f_n)_{n\ge 0}\in \mathcal {M}\) and \(B\in \mathcal {F}\) such that \(S(f)\le \chi _B\), then there exists a positive constant \(C_\phi \) independent of f and B such that
The proof of Lemma 15 is similar to the proof of Lemma 5.3 in [24]. Its proof is left to the reader.
If we consider the case of \( \phi (t)=t^\alpha \) for \(\alpha >0\) and \(t\in [0,\infty )\), the fractional integral operator \(I_\alpha \) has the following property.
Lemma 16
([23]) Let \(\alpha >0\) and \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular. If \(f=(f_n)_{n\ge 0}\in \mathcal {M}\) and \(B\in \mathcal {F}\) such that \(S(f)\le \chi _B\), then there exists a positive constant \(C_\alpha \) independent of f and B such that
It is noteworthy that the case of \(\alpha \ge 1\) in Lemma 16 is not included in Lemma 15.
Lemma 17
Let \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular, \(0<q\le \infty \), \(\phi \) be an Orlicz function of upper type \(p\in (0,1)\) and let b be a slowly varying function. Suppose that a is a \((2,\Phi ,\infty )\)-atom as in Definition 4. If there exists \(\Psi \) such that \( \Psi ^{-1}(t)\ge \Phi ^{-1}(t) \cdot \phi (1/t) \) for any \(t\ge 1\), then we have
where \(\tau \) is the stopping time associated with \((2,\Phi ,\infty )\)-atom a and \(C_\phi \) is the same constant as in Lemma 15.
Proof
From the definition of the \((2,\Phi ,\infty )\)-atom a, we have
This means
By using Lemma 15, we can obtain that
Then, by the condition \( \Psi ^{-1}(t)\ge \Phi ^{-1}(t) \cdot \phi (1/t) \) for any \(t\ge 1\), we have
Now, let
Then \((\lambda _n)_{n\ge 0}\) is a non-negative, non-decreasing and adapted sequence and, similarly to (4.7), \(S_{n+1}(I_\phi a) \le \lambda _n\). Hence, we obtain
where \(C_\phi \) is the same constant as in Lemma 15. The proof is complete. \(\square \)
Following the same proof as in Lemma 17, we obtain the following conclusion via Lemma 16.
Lemma 18
Let \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular, \(0<q\le \infty \), \(\alpha >0\) and let b be a slowly varying function. Suppose that a is a \((2,\Phi ,\infty )\)-atom as in Definition 4, where \(\Phi \) is an Orlicz function. If there exists \(\Psi \) such that \( \Phi ^{-1}(t) \le \Psi ^{-1}(t) \cdot t^\alpha \) for any \(t\ge 1\), then we have
where \(\tau \) is the stopping time associated with \((2,\Phi ,\infty )\)-atom a and \(C_\alpha \) is the same constant as in Lemma 16.
When we consider \(\Phi (t)=t^p\) and \(\Psi (t)=t^r\) in Lemma 18, one can see that
Corollary 12
Let \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular, \(0<q\le \infty \), \(\alpha >0\) and let b be a slowly varying function. Suppose that a is a \((2,p,\infty )\)-atom for \(0<p<\infty \). If there exists r such that \( \alpha \ge \frac{1}{p}- \frac{1}{r}\), then we have
where \(\tau \) is the stopping time associated with \((2,p,\infty )\)-atom a and \(C_\alpha \) is the same constant as in Lemma 16.
We now prove the boundedness of generalized fractional integrals on Orlicz-Lorentz-Karamata Hardy martingale spaces via atomic decomposition. Recall that all the spaces \(H_{\Phi ,q,b}^S\), \(\mathcal {Q}_{\Phi ,q,b}\), \(\mathcal {P}_{\Phi ,q,b}\), \(H_{\Phi ,q,b}^M\) and \(H_{\Phi ,q,b}^s\) are equivalent if \(\{\mathcal {F}_n\}_{n\ge 0}\) is regular (see Theorem 9).
Theorem 14
Let \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular, \( 0 < s\le \min \{1,q\} \) and b be a slowly varying function. Let \( \Psi \) be an Orlicz function of lower type \(p_\Psi ^-\) and upper type \(p_\Psi ^+\) satisfying \(s<p_\Psi ^- \le p_\Psi ^+<\infty \) and let \( \Phi \) be an Orlicz function of lower type \(p_\Phi ^-\) and upper type \(p_\Phi ^+\) satisfying \(0<p_\Phi ^- \le p_\Phi ^+<\infty \). If there exists an Orlicz function \(\phi \) with upper type \(p\in (0,1)\) such that \( \Phi ^{-1} (t) \cdot \phi (1/t) \le \Psi ^{-1} (t) \) for any \( t \ge 1 \), then there exists a constant C such that
for all \(f\in \mathcal {Q}_{\Phi ,s,b}\).
Proof
Let \(f\in \mathcal {Q}_{\Phi ,s,b}\). According to Theorem 8, there exist a sequence \((a^k)_{k\in \mathbb {Z}}\) of \((2,\Phi ,\infty )\)-atoms and a sequence of real numbers
(\(\tau _k\) is the stopping time associated with \(a^k\)) such that for all \(n\ge 0\),
and
Combining this with Lemma 17 and Theorem 2, we can finish the proof of Theorem 14. Indeed, since \( 0 < s \le \min \{1,q\} \) and \( s<p_{\Psi }^- \), we have
This completes the proof of the theorem. \(\square \)
Based on the argument similar to the proof of Theorem 14 and using Lemma 18, we obtain the following result.
Theorem 15
Let \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular, \( 0 < s\le \min \{1,q\} \) and b be a slowly varying function. Let \( \Psi \) be an Orlicz function of lower type \(p_\Psi ^-\) and upper type \(p_\Psi ^+\) satisfying \(s<p_\Psi ^- \le p_\Psi ^+<\infty \) and let \( \Phi \) be an Orlicz function of lower type \(p_\Phi ^-\) and upper type \(p_\Phi ^+\) satisfying \(0<p_\Phi ^- \le p_\Phi ^+<\infty \). If there exists \(\alpha >0\) such that \( \Phi ^{-1} (t) \le \Psi ^{-1} (t) \cdot t^\alpha \) for any \( t \ge 1 \), then there exists a constant C such that
for all \(f\in \mathcal {Q}_{\Phi ,s,b}\).
As an application of Theorem 14, we obtain the following result:
Corollary 13
Let \( \{\mathcal {F}_n\}_{ n \ge 0} \) be regular, \( 0 < s\le \min \{1,q\} \), \( \Psi \) be an Orlicz function of lower type \(p_\Psi ^-\) and upper type \(p_\Psi ^+\) satisfying \(s<p_\Psi ^- \le p_\Psi ^+<\infty \) and let \( \Phi \) be an Orlicz function of lower type \(p_\Phi ^-\) and upper type \(p_\Phi ^+\) satisfying \(0<p_\Phi ^- \le p_\Phi ^+<\infty \). If there exists an Orlicz function \(\phi \) with upper type \(p\in (0,1)\) such that \( \Phi ^{-1} (t) \cdot \phi (1/t) \le \Psi ^{-1} (t) \) for any \( t \ge 1 \), then there exists a constant C such that
for all \(f\in \mathcal {Q}_{\Phi ,s}\).
Proof
Let \(b\equiv 1\) in Theorem 14. \(\square \)
The next corollary follows from Theorem 15.
Corollary 14
Suppose that \( \{\mathcal {F}_n\}_{ n \ge 0} \) is regular, \(0<q_1\le \min \{1,q_2, p_2\}\), \(0<p_1<p_2<\infty \), \(\alpha \ge {1}/{p_1}- {1}/{p_2}\) and b is a slowly varying function. Then there exists a constant C such that
for all \(f\in \mathcal {Q}_{p_1,q_1,b}\).
Remark 15
Note that this result was proved in [31] and [12] for \(1<p_1<p_2<\infty \). Moreover, the slowly varying function b is not necessarily non-decreasing. Thus, Corollary 14 improves and completes Theorem 4.4 in [39].
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Acknowledgements
The authors would like to thank the referees for carefully reading the paper and valuable comments. Zhiwei Hao is supported by the NSFC (No. 11801001) and Hunan Provincial Natural Science Foundation (No. 2022JJ40145), Libo Li is supported by the NSFC (No. 12101223) and Hunan Provincial Natural Science Foundation (No. 2022JJ40146).
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Hao, Z., Li, L., Long, L. et al. Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators. Fract Calc Appl Anal 27, 554–615 (2024). https://doi.org/10.1007/s13540-024-00259-3
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DOI: https://doi.org/10.1007/s13540-024-00259-3
Keywords
- Martingale
- Orlicz-Lorentz-Karamata spaces
- Doob’s inequality
- Atomic decompositions
- Duality
- Fractional integrals