Characterizations of variable martingale Hardy spaces via maximal functions

1 Introduction

For a measurable function p(⋅), the variable Lebesgue space L_p(⋅) consists of all measurable functions f for which ∫01 ∣f(x)∣^p(x)dx < ∞. If p(⋅) is a constant, we get back the usual L_p space. This topic needs essentially new ideas and is investigated very intensively in the literature nowadays (see e.g. Cruz-Uribe and Fiorenza [6], Diening et al. [7], Kokilashvili et al. [16, 17], Kováčik and Rákosník [18], Cruz-Uribe et al. [4, 3, 2], Nakai and Sawano [22, 31], Kempka and Vybíral [15], Rafeiro et al. [24, 27], Samko [8, 28, 29], Jiao et al. [11, 12, 14], Yan et al. [40], Liu et al. [19, 20]). Interest in the variable Lebesgue spaces has increased since the 1990s because of their use in a variety of applications (see the references in Jiao et al. [11]).

Usually, we suppose that p(⋅) or 1/p(⋅) satisfy the log-Hölder continuity condition. The classical Hardy-Littlewood maximal operator is bounded on the variable L_p(⋅) spaces if the exponent function p(⋅) is log-Hölder continuous and 1 < p₋ ≤ p₊ < ∞, where p₋ denotes the infimum and p₊ the supremum of p(⋅) (see for example Cruz-Uribe et.al [4] and Nekvinda [23]). Moreover, under the log-Hölder continuity condition of 1/p(⋅), the maximal operator is bounded on L_p(⋅) when 1 < p₋ ≤ p₊ ≤ ∞ (see e.g. Cruz-Uribe and Fiorenza [6] and Diening et al. [7]). The fractional integral operator was investigated in Ephremidze et al. [8], Rafeiro and Samko [27, 25, 26 30] and, for martingales, in Hao et al. [9, 13].

The boundedness result for the usual dyadic (or martingale) maximal operator on the L_p(⋅) spaces was proved in [11, 12] if 1 < p₋ ≤ p₊ < ∞. In [36], we generalized this result to 1 < p₋ ≤ p₊ ≤ ∞. In [11, 35], we investigated four more dyadic maximal operators and show that they are bounded on L_p(⋅) if 1 < p₋ ≤ p₊ < ∞. The boundedness of these operators was the key point in proving the boundedness of the maximal Fejér, Cesàro and Riesz operators of the Walsh-Fourier series from the variable Hardy space H_p(⋅) to L_p(⋅) (see [11, 33]).

Nakai and Sawano [22] first introduced the Hardy space H_p(⋅)(ℝ) with a variable exponent p(⋅) and established the atomic decompositions. Independently, Cruz-Uribe and Wang [5] also investigated the variable Hardy space H_p(⋅)(ℝ). Sawano [31] improved the results in [22]. Ho [10] studied weighted Hardy spaces with variable exponents. Recently, Yan et al. [40] introduced the variable weak Hardy space H_p(⋅),∞(ℝ) and characterized these spaces via radial maximal functions. The Hardy-Lorentz spaces H_{p(⋅), q}(ℝ) were investigated by Jiao et al. in [14]. Similar results for the anisotropic Hardy spaces H_p(⋅)(ℝ) and H_{p(⋅), q}(ℝ) can be found in Liu et al. [19, 20]. Martingale Musielak–Orlicz Hardy spaces were investigated in Xie et al. [37, 38, 39]. Very recently, these results are generalized for martingale Hardy spaces with variable exponent in Jiao et al. [11].

In this paper, we introduce a common generalization of the usual dyadic and the other four maximal operators, denoted by U_γ,s, where γ and s are positive parameters. We prove that if 1/p(⋅) satisfy the log-Hölder continuity condition, 1p−−1p+<γ+s and 1 < p₋ ≤ p₊ ≤ ∞, then U_γ,s is bounded on L_p(⋅). We also verify that under the same conditions, ∥U_γ,sf∥_Lp(⋅) is equivalent to ∥f∥_Hp(⋅) and ∥U_γ,sf∥_Lp(⋅),q is equivalent to ∥f∥_Hp(⋅),q, where H_p(⋅) and H_{p(⋅), q} denote the variable Hardy and Hardy-Lorentz spaces. Moreover, we show with a counterexample that the condition 1p−−1p+<γ+s is important, the results do not hold without this condition.

2 Variable Lebesgue and Lorentz spaces

In this section, we recall some basic notations on variable Lebesgue spaces and give some elementary and necessary facts about these spaces. Our main references are Cruz-Uribe and Fiorenza [6] and Diening et al. [7].

For a constant p, the L_p space is equipped with the quasi-norm

with the usual modification for p = ∞. Here we integrate with respect to the Lebesgue measure λ.

We are going to generalize these spaces. A measurable function p(⋅) : [0, 1) → (0, ∞] is called a variable exponent. For a measurable set A ⊂ [0, 1), we denote

and for convenience

Denote by 𝓟 the collection of all variable exponents p(⋅) such that

In what follows, we use the symbol

We define the modular functional by

where Ω_∞ = {x ∈ [0, 1) : p(x) = ∞}. The variable Lebesgue space L_p(⋅) is the collection of all measurable functions f for which there exists ν > 0 such that

This becomes a quasi-Banach function space when it is equipped with the quasi-norm

If p(⋅) = p is a constant, then we get back the definition of the usual L_p spaces. For any f ∈ L_p(⋅), we have ρ(f) ≤ 1 if and only if ∥f∥_p(⋅) ≤ 1 (see Cruz-Uribe and Fiorenza [6]). It is known that ∥ν f∥_p(⋅) = ∣ν∣∥f∥_p(⋅),

and

where p(⋅) ∈ 𝓟, s ∈ (0, ∞), ρ ∈ ℂ and f, g ∈ L_p(⋅). Details can be found in the monographs Cruz-Uribe and Fiorenza [6] and Diening et al. [7]. The variable exponent p′(⋅) is defined pointwise by

The next two lemmas are well known, see Cruz-Uribe and Fiorenza [6] or Diening et al. [7].

3 Variable martingale Hardy spaces

Let 𝓕_n be the σ-algebra

where σ(𝓗) denotes the σ-algebra generated by an arbitrary set system 𝓗. The conditional expectation operators relative to 𝓕_n are denoted by E_n. An integrable sequence f = (f_n)_n∈ℕ is said to be a martingale if f_n is 𝓕_n-measurable for all n ∈ ℕ and E_n f_m = f_n in case n ≤ m. Martingales with respect to (𝓕_n, n ∈ ℕ) are called dyadic martingales. It is easy to show (see e.g. Weisz [34]) that the sequence (𝓕_n, n ∈ ℕ) is regular, i.e., f_n ≤ Rf_{n-1} for all non-negative dyadic martingales.

For a dyadic martingale f = (f_n)_n∈ℕ, the maximal function is defined by

Now we can define the variable martingale Hardy spaces by

The variable martingale Hardy-Lorentz spaces can be defined similarly:

The Hardy spaces can also be defined via equivalent norms. In [11], we have shown that the Hardy spaces defined the square function and the conditional square function are equivalent. In this paper, we give other equivalent characterization of the variable Hardy spaces via new maximal functions.

The atomic decomposition is a useful characterization of the Hardy spaces. A measurable function a is called a p(⋅)-atom if there exists a stopping time τ such that

1. E_n(a) = 0 for all n ≤ τ,

2. ∥M(a)∥∞≤χτ<∞p(⋅)−1.

The atomic decomposition of the spaces H_p(⋅) and H_{p(⋅), q} were proved in Jiao et al. [11, 12]. The classical case can be found in [34].

4 The boundedness of maximal operators on L_p(⋅)

The following result was proved in Jiao et al. [11, 12] if p₊ < ∞ and by the author [36] if p₊ ≤ ∞. For the boundedness of the classical Hardy-Littlewood maximal operator see e.g. Cruz-Uribe and Fiorenza [6] and Diening et al. [7].