Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces

: In this paper


Introduction
There is a greatly interested to study variable exponent spaces and operators with variable parameters in last two decade. Some researches are monographs about Lebesgue spaces with variable exponents, for example, [1,2]. Let p(·) be a measurable function : Ω → [1, ∞). We suppose that where p − = ess inf x∈Ω p(x), p + = ess sup x∈Ω p(x). We let L p(·) (Ω) be the set of functions f such that It is a Banach space equipped with the norm f p(·) = inf η > 0 : ρ p(·) (f /η) 1 , we denote the conjugate exponent by p (x) = p(x) p(x)−1 for x ∈ Ω.
Let M be the Hardy-Littlewood maximal operator, i.e., for f ∈ L 1 loc (R n ) where the supremum is taken over all cubes Q containing x, and the |Q| is the Lebesgue measure of Q ⊂ R n . Let B(Ω) be the set such that M is bounded on L p(·) (Ω). Diening [3] proved that p(·) ∈ B(Ω) if p(·) ∈ LH(Ω), i.e M is bounded on L p(·) (Ω). When Ω is unbounded to see [4]. Morrey spaces play an important role in study of local properties of solutions of partial differential equations. These spaces were introduced by Morrey [11] in 1938. The norm of Morrey space is defined as follows: where Q is the cube with the center x and with side-length r and its sides parallel to the coordinate axes, 0 < λ < n and 1 p < ∞. Variable exponent Morrey spaces were firstly introduced in [13] and also studied in [12]. Generalized variable exponent Morrey spaces L p(·),ω (Ω) were introduced in [6]. They proved the boundedness of maximal, singular, potential operator in variable exponent Morrey spaces L p(·),ω (Ω). When ω(x, r) = r λ(x)−n p(x) , the L p(·),ω (Ω) are the variable exponent Morrey spaces L p(·),λ(·) (Ω) in [8].
The generalized variable exponent Morrey space where we assume that inf x∈Ω,r>0 It is the variable Morrey space L p(·),λ(·) (Ω) under the is a measurable function on Ω with values in [0, n]. When p is constant, the norm is defined by In this paper we mainly consider the boundedness of intrinsic square functions in variable Morrey spaces. Let The corresponding Lusin area integral S β is defined by The Littlewood-Paley g-function and g * λ -function are defined respectively by Let ψ t (x) = t −n ψ(x/t) and ψ ∈ C ∞ 0 (R n ) be real, radial, supported in {x : |x| ≤ 1} and for all ξ = 0, where ψ denotes the Fourier transform of ψ. The continuous square functions S ψ,β and Littlewood-Paley g-function g ψ are defined by For 0 < α ≤ 1, let C α be the family of function φ : R n → R such that it is supported in {x : |x| ≤ 1}, φ(x)dx = 0, and for all x and x In [16], the intrinsic square function is defined by The intrinsic Littlewood-Paley g-function and the intrinsic g * λ -function are defined respectively by We recall several properties of the intrinsic square function, the proofs of them to see [16,17].
(1) G α is of weak type (1,1): (2) If β 1, then for all x ∈ R n , (3) If S is anyone the Littlewood-Paley operators defined above, then where the constant C is independent of f and x; (4) The function G α and g α are pointwise comparable with comparability constants only depending on α and n.
dx and the supremum is taken over all cube Q whose sides parallel to the coordinate axes in R n . The commutators generated by BMO function b and intrinsic square functions are respectively defined by In order to study the commutators, we need the following properties of BMO. For b ∈ BMO and 1 < p < ∞, we get For all nonnegative integers l, we obtain Wang [19] has estimated this class operator for ω is increasing and there is a constant D, 1 D < 2 n , such that, for any r > 0, ω(2Q) Dω(Q), where ω(Q) = ω(x, r), Q is the cube with the center x and with side-length r and its sides parallel to the coordinate axes. We can impose the following condition on ω(x, r) as in [10].
Assume that there is a constant C such that, for any x ∈ R n and for any r > 0, Theorem 1.1. Assume that ω(x, r) satisfy the conditions (8) and (9). Then (i) there is a constant C p > 0 such that for f ∈ L p,ω and 1 < p < ∞; (ii) there is a constant C p > 0 such that for any σ > 0 and any cube Q, for f ∈ L p,ω and 1 < p < ∞. Based on Theorem 1.1, (10) holds for the classical Calderón-Zygmund operators along with the Littlewood-Paley technique in [15,16,17] (8) and (9). Then for 1 < p < ∞, there is a constant C p > 0 such that (10) holds for T.

Preliminaries
A nonnegative locally integrable function w belongs to A p (p > 1) if where p is the conjugate index of p i.e. 1/p + 1/p = 1. Lemma 2.1. [18] The intrinsic square function G α , the Lusin area integral S β , the Littlewood-Paley function g, the continuous square functions S ψ and g ψ are bounded on L p (w) for w ∈ A p (1 < p < ∞).

An important extrapolation theorem was introduced in [5].
Lemma 2.4. Let f and g be non-negative and measurable functions, assume that there exists a constant C such that holds for some 1 < q < ∞ and for every w ∈ A q . Then for p(·) ∈ LH(Ω).
We can obtain the following conclusion by Lemma 2.1 and Lemma 2.4.
The Littlewood-Paley g * λ -function has been discussed in [5]. For commutators, we have the following theorem by using Lemma 2.2 and Lemma 2.4.

Proof of the Main Results
Proof of Theorem 1.2: By (4) and (5), it is enough to prove Theorem 1.1 for G α (f ). (i) Let Q be a cube of R n , with the center x 0 and side-length r. Decompose f = f χ 2Q + f χ (2Q) c and denote f χ 2Q by f 1 and f χ (2Q) c by f 2 , χ 2Q denotes the characteristic function of set 2Q. Since G α (f ) is a sublinear, we have Lemma 2.1 and (8) imply For any x ∈ Q, (y, t) ∈ Γ(x) and z ∈ (2 l+1 Q \ 2 l Q) ∩ Q(y, t), l ∈ Z + , then

By the Minkowski inequality and Hölder inequality, we have
It is easy to know that by (8) and (9). So we get ω(x 0 , r) r n . Hence Combining the estimate for I 1 and I 2 , we obtain the (10).
(ii) For σ > 0, by (3) and (10), we get (11). 2 In order to prove 1.2 we need the following estimate. Proposition 3.1. Let 0 < θ 1. Assume that ω satisfies 1/C ω(x, t)/ω(x, r) C, f or r t 2r Then for 1 p < ∞ there is a constant C > 0 such that for f ∈ L p,ω , any cube Q, β and N large enough, where G N is the grand maximal function .
The following important weight norm inequality is proved in [15]: That is for 0 < p < ∞, there exits a "Maximal operator" M such that (14) is valid for f ∈ L p (R n ), any nonnegative V ∈ L 1 loc (R n ), appropriate ψ and large enough β and N with a constant C which is independent V and f. For every p, we can take M = M k (k > 1); if k = 1, (14) holds for p ∈ (0, 2).
The Proposition 3.1 tells us for large enough β and N.
It is well known that for all x ∈ R n [22, pp. 67-68], Combining this with (15) and (16), we have This estimate along with Theorem 1.1 and Lemma 2.7, (10) yields for T. 2 Proof of Theorem 1.2 : (i) As before, we set f = f 1 + f 2 , f 1 = f χ 2Q . From the definition of g * λ,α , we can easily deduce that By Theorem 1.1, we know that I 0 C f p,ω . We can get the following estimate for I 1 j from [19], i.e.
The estimate of I (2) j is similar to I 2 , (ii)For σ > 0, by Lemma 2.3 and the estimate of I (2) j , the (11) is valid to g * λ,α . 2 Proof of Theorem 1.3 : Fix a cube Q of R n whose center is x 0 and edges have length r. Decompose f = f χ 2Q + f χ (2Q) c and denote f χ 2Q by f 1 and f χ (2Q) c by f 2 .
By Lemma 2.2 we get In the proof of Theorem 1.1, we get that for any x ∈ Q, And by the (6), we obtain Next we deal with II 21 . It is similar to |G α (f 2 )(x)|, Next we estimate G α (f 2 ). It is similar to the second estimate in proof (i) of Theorem 1.1. We have Choosing ε > n p− , by the Hölder inequality of variable exponent and Lemma 2 we have s ε+1 f L p(·) ( Q(x0,s)) |x 0 − z| −n+ε L p (·) ( Q(x0,s)) ds C t s − n p(x) −1 f L p(·) ( Q(x0,s)) ds.