New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

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The analogous boundedness result for the martingale maximal operator on the L p(·) spaces was proved in [16,19] if 1 < p − ≤ p + < ∞. Later we generalized this result to 1 < p − ≤ p + ≤ ∞ in [45]. In [16,43,44,46], we investigated more general maximal operators for dyadic martingales and verified that they are bounded on L p(·) if 1 < p − ≤ p + < ∞. These operators were the key points in the proof of the boundedness of the maximal Fejér operator of the Walsh-Fourier series from the variable Hardy space H p(·) to L p(·) (see [16,40,46]). In this paper we generalize these operators and introduce the general fractional maximal operator for more general martingales.
Nakai and Sawano [29] first introduced the variable Hardy spaces H p(·) (R). Independently, Cruz-Uribe and Wang [4] also investigated the spaces H p(·) (R). Sawano [37] improved the results in [29]. Ho [14] studied weighted Hardy spaces with variable exponents. Yan et al. [50] introduced the variable weak Hardy spaces 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 [21]. Similar results for the anisotropic Hardy spaces H p(·) (R) and H p(·),q (R) can be found in Liu et al. [26,27]. The theory of martingale Hardy spaces with variable exponents was started with the paper Jiao et al. [19]. Some years later, we have systematically studied and developed this theory in [16]. The applications of variable martingale Hardy spaces to Fourier analysis were investigated in [16] while its applications to stochastic integrals in [18]. Martingale Musielak-Orlicz Hardy spaces were considered in Xie et al. [47][48][49].
In this paper, we generalize all the maximal operators mentioned above and introduce a new, general fractional maximal operator M γ,s,α for so called Vilenkin martingales (γ , s > 0, 0 < α ≤ 1). This maximal operator is larger than the Doob maximal operator or the original fractional maximal operator. We generalize the boundedness of the Doob and fractional maximal operators and prove that M γ,s,α is bounded from L q(·) to L p(·) if 1/ p(·) is log-Hölder continuous, 1 < q − ≤ q + ≤ 1/α, 1/ p(·) = 1/q(·) − α and 1/ p − − 1/ p + < γ + s. Under the same conditions, we obtain also the boundedness of M γ,s,α from H q(·) to L p(·) . Moreover, we show that for α = 0, M γ,s,0 f L p(·) is equivalent to f H p(·) , where H p(·) denotes the variable martingale Hardy spaces. Finally, we show that the condition 1 p − − 1 p + < γ + s is important.

Variable Lebesgue 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 [5] and Diening et al. [8].
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. For the integral of f , we will use both symbols and for convenience Denote by P the collection of all variable exponents p(·) such that To introduce the variable Lebesgue spaces, let where Ω ∞ = {x ∈ [0, 1) : p(x) = ∞}. We denote the set [0, 1) \ Ω ∞ also by Ω c ∞ . The variable Lebesgue space L p(·) is the collection of all measurable functions f for which there exists ν > 0 such that We equip L p(·) with the (quasi)-norm If p(·) = p is a constant, then we get back the definition of the usual Lebesgue spaces L p . For any f ∈ L p(·) , we have ρ( f ) ≤ 1 if and only if f p(·) ≤ 1. It is known that ν f p(·) = |ν| f p(·) and where p(·) ∈ P, s ∈ (0, ∞), ν ∈ C and f ∈ L p(·) . Details can be found in the monographs Cruz-Uribe and Fiorenza [5] and Diening et al. [8]. The variable exponent p (·) is defined pointwise by The next lemma is well known, see Cruz-Uribe and Fiorenza [5] or Diening et al. [8]. Moreover, where ∼ denotes the equivalence of the numbers.
We denote by C log the set of all functions p(·) ∈ P satisfying the so-called log-Hölder continuous condition, namely, there exists a positive constant C log ( p) such that, for any x, y ∈ [0, 1), . (2.1)

Remark 1
There exist a lot of functions p(·) satisfying (2.1). For concrete examples we mention the function a +cx for parameters a and c such that the function is positive (x ∈ [0, 1)). All positive Lipschitz functions with order 0 < η ≤ 1 also satisfy (2.1). Under the condition 0 < p − ≤ p + < ∞, p(·) ∈ C log if and only if 1/ p(·) ∈ C log .
In this paper the constants C are absolute constants and the constants C p(·) are depending only on p(·) and may denote different constants in different contexts. For two positive numbers A and B, we use also the notation A B, which means that there exists a constant C such that A ≤ C B.

Variable martingale Hardy spaces
Let ( p n , n ∈ N) be a sequence of natural numbers with entries at least 2. We always suppose that the sequence ( p n ) is bounded. Let Introduce the notations P 0 = 1 and Let F n be the σ -algebra where σ (H) denotes the σ -algebra generated by an arbitrary set system H. By a Vilenkin interval we mean one of the form [k P −1 n , (k + 1)P −1 n ) for some k, n ∈ N, 0 ≤ k < P n . 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 nmeasurable 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 Vilenkin martingales. Vilenkin martingales were introduced in a great number of papers, such as Gát and Goginava [10][11][12], Persson and Tephnadze [31][32][33][34] and Simon [38,39]. It is easy to show (see e.g. Weisz [41]) that the sequence (F n , n ∈ N) is regular, i.e., f n ≤ p f n−1 for all non-negative Vilenkin martingales with p just defined in (3.1).
For a Vilenkin martingale f = ( f n ) n∈N , the Doob maximal function is defined by For a fixed x ∈ [0, 1) and n ∈ N, let us denote the unique Vilenkin interval [k P −1 n , (k + 1)P −1 n ) which contains x by I n (x). Then it is easy to see that If f ∈ L 1 , then we can replace f n by f in the integral. Now we can define the variable martingale Hardy spaces by The Hardy spaces can also be defined via equivalent norms. For a martingale f = ( f n ) n≥0 , let denote the martingale differences, where f −1 := 0. The square function and the conditional square function of f are defined by We have shown the following theorem in [16]. Then If in addition 1 < p − ≤ p + < ∞, then H p(·) ∼ L p(·) .
In this paper, we will give more equivalent characterizations for the variable Hardy spaces using the 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 Vilenkin interval I such that (a) the support of a is contained in I , The atomic decompositions of the spaces H p(·) were proved in Jiao et al. [16,19]. The classical case can be found in [41,42].
almost everywhere for every n ∈ N, where (a k,K ,κ ) k∈Z,K ,κ∈N is a sequence of p(·)-atoms associated with the Vilenkin intervals (I k,K ,κ ) k∈Z,K ,κ∈N ⊂ F K , which are disjoint for fixed k, and μ k,K ,κ = 3 · 2 k χ I k,K ,κ p(·) . Moreover, where 0 < t ≤ p := min{ p − , 1} is fixed and the infimum is taken over all decompositions of f as above.

The boundedness of fractional maximal operators on L p(·)
Let 0 ≤ α ≤ 1. The original fractional maximal operator generates the Doob maximal operator M. Indeed, for α = 0, we get back the definition of M because of (3.2). If f is an integrable function, then we can replace f n by f in the definition of M α . The following result was proved for the classical fractional maximal operator and for L p(·) (R) spaces in Capone [2], Cruz-Uribe and Fiorenza [5] or Kokilashvili and Meskhi [22] and for more general martingales in Jiao et al. [16,19] if p + < ∞, α = 0 and by the author [45] if p + ≤ ∞, α = 0.
We will generalize the preceding martingale fractional maximal operator. Every point x ∈ [0, 1) can be written in the following way: If there are two different forms, choose the one for which lim k→∞ x k = 0. The so called Vilenkin addition is defined by Given two Vilenkin intervals I and J , let I+J := {x+y : x ∈ I , y ∈ J }. For a Vilenkin interval I with length P −1 n , i, j, n ∈ N, l = 0, . . . , p j − 1, let us use the notation Parallel, we denote I n (x) l, j,i := (I n (x)) l, j,i . Recall that I n (x) is a Vilenkin interval such that I n (x) ∈ F n and x ∈ I n (x). Let γ and s be two positive constants. Now we introduce four new fractional maximal functions and a common generalization of these maximal functions. For a martingale f = ( f n ) n≥0 , let Here γ,s,α and M (2) γ,s,α are equivalent to M α , more exactly, and so Theorem 3 holds also for these two operators. The third fractional maximal operator is defined by . We define our last fractional maximal operator by The maximal operators M γ,s,α and M (4) γ,s,α cannot be estimated by M α from above pointwise. For α = 0, we considered these two maximal operators in [46] and used them to verify some boundedness and convergence results for the Fejér means of the Vilenkin-Fourier series. We proved there that they are bounded on L p(·) if 1/ p(·) ∈ C log and 1 < p − ≤ p + < ∞. In this paper, we will generalize this result for a more general operator, for 1 < p − ≤ p + ≤ ∞ and 0 < α ≤ 1. The next general fractional maximal operator generalizes our operators M Of course, if f ∈ L 1 , then we can write again in the definition f instead of f n . For α = 0, we omit simply α and write M γ,s ( f ) and M( f ). Obviously, we obtain from It is easy to see that In this section, we will use the next lemma proved in [45]. A first version of this lemma can be found in Jiao et al. [16,19].
We will also use the following simple lemma.

Lemma 5
Suppose that 0 ≤ α < 1 and 1 ≤ r ≤ 1/α. Then for all intervals I ⊂ [0, 1) and all functions f ∈ L r with f r ≤ 1, The first term is bounded by 1 because, by Hölder's inequality, which finishes the proof of the lemma.
Our first boundedness result on the general fractional operator with constant p and q can be read as follows.
Proof For α = 0, the theorem can be proved similarly to Corollary 4 in [46], so we omit the details. For 0 < α < 1, we first note that Since p ≥ q, we get by convexity that Observe that p(1 − αq)/q = 1. Now we can apply Lemma 5 to obtain which finishes the proof.
Now we generalize this theorem as well as Theorem 3 to variable Lebesgue spaces as follows. The methods used in the proof of Theorems 3 and 4 do not work now, so we need new ideas. and then M γ,s,α ( f ) p(·) f q(·) ( f ∈ L q(·) ).

F. Weisz
Proof It is enough to show the theorem for f q(·) = 1 and for non-negative functions.
We decompose f as f 1 + f 2 , where Then f i q(·) ≤ 1 and ρ( f i ) ≤ 1, i = 1, 2. Similarly to (4.2), Since ρ is convex, we get that where 8ηC s C γ p < β < 1 and β is given in (2.2). For a fixed k, n, let us denote by Λ k,n those triples (l, j, i) for which 0 ≤ j ≤ n, 0 ≤ i ≤ n, l = 0, . . . , p j − 1 and Then It is easy to see that We denote by I k,n,l, j,i,1 (resp. I k,n,l, j,i,2 ) those points x ∈ I k,n for which Let u(x) := p(x)/ p 0 > 1 for some 1 < p 0 < q − < p − , where we will choose p 0 near to 1 later. By convexity and by the disjointness of the sets I k,n for a fixed n, we conclude Equality We use Lemma 5 with r = q − (I l, j,i k,n ) to see that Hence Here we have also used that β < 1 and u − (I we have that f q(·)/ p 0 ≤ 1 and we can apply Lemma 4 with p(·) = q(·)/ p 0 . By Lemma 4 and Theorem 4, we conclude Choosing 0 < γ 0 < γ and 0 < r < s + γ 0 , we obtain where we can choose η such that 8ηC γ −γ 0 C γ 0 +s−r p ≤ β p − . By (4.7), Hölder's inequality and by the fact that λ(I l, j,i Consequently, Recall that supp f 1 ⊂ Ω c ∞ and f 1 > 1 or f 1 = 0. Since u(x) > u − (I l, j,i k,n ) on I k,n,l, j,i,2 , q − (I l, j,i k,n )/ p 0 ≤ q(t) for all t ∈ I l, j,i k,n and we can see that For fixed k, n, let J j denote the Vilenkin interval with length P −1 j and I k,n ⊂ J j . Then I l, j,i k,n ⊂ J j+ l P −1 j+1 = J j . Inequality (2.2) implies that, for any x ∈ I k,n , (4.13) We can choose γ 0 , r and p 0 such that (4.14) After an easy calculation, we can see that By this and (4.13), We estimate (4.12) further by 18 F. Weisz whenever (4.14) holds. Since r can be arbitrarily near to s + γ 0 , γ 0 to γ and p 0 to 1, (4.14) gives (4.4). Now, we consider the second term of (4.5). We may suppose that Ω ∞ has positive measure and so p + = ∞. Let x ∈ Ω ∞ be fixed. Since This means that x ∈ I k,n and so p + (I k,n ) = p + = ∞. Let J j denote the Vilenkin interval with length P −1 j such that I k,n ⊂ J j ( j = 0, . . . , n). Then p + (J j ) = ∞ and Let us choose v ∈ R such that 2 v ≥ P n and v ≥ p − (I l, j,i k,n ) for all j = 0, . . . , n, i = j, . . . , n and l = 0, . . . , p j − 1.
We can see as in (4.9) and (4.10) that .
In Theorem 7 we will show that condition (4.4) is important in Theorem 5, the result is not true without this condition.

If (4.3) and (4.4) hold, then
Proof According to Theorem 2, f can be written as where (a k,K ,κ ) k∈Z,K ,κ∈N is a sequence of q(·)-atoms associated with the Vilenkin intervals (I k,K ,κ ) k∈Z,K ,κ∈N ⊂ F K , which are disjoint for fixed k, and 0 < t < q ≤ p. Then We first estimate Z 1 . Since 0 < t < q ≤ p ≤ 1, we have By Lemma 1, we may choose g ∈ L ( p(·) t ) with norm less than 1 such that Choose r > 1 such that rt > max( p + , 1/(1 − α)). By Hölder's inequality we get that Theorem 5 yields that M γ,s,α is bounded from L v to L rt , where v is defined by Using the definition of μ k,K ,κ , we estimate Z 1 further by .
Note that rt > p + ≥ q + implies that (q(·)/t) > r . This and equality (4.3) mean that and so we can apply Theorem 5 to obtain that M αtr is bounded from L ( p(·)/t) r to L (q(·)/t) r . More precisely,

F. Weisz
Consequently, Now we estimate Z 2 . The operator M γ,s,α can also be written in the following way: where I ∈ F n is a Vilenkin interval. We suppose that x / ∈ I k,K ,κ . Since Thus we can suppose that i > K , and so n ≥ m > K . If x / ∈ I k,K ,κ , x ∈ I and j ≥ K , then I l, j,i ∩ I k,K ,κ = ∅. Therefore we can suppose that j < K . Similarly, if then I l, j,i ∩ I k,K ,κ = ∅, so we may assume that x ∈ I k,K ,κ+ l P −1 j+1 = I l, j,K k,K ,κ . Therefore, for x / ∈ I k,K ,κ , Since the function x → x2 −γ x is bounded, we obtain that Consequently, where 0 < t < q ≤ p. Let us choose max(1, p + ) < r < ∞. By Lemma 1, we can find a g ∈ L ( p(·) t ) with g ( p(·) t ) ≤ 1 such that dx.