Intrinsic square functions and commutators on Morrey‐Herz spaces with variable exponents

In this article, we will study the boundedness of intrinsic square functions on the Morrey‐Herz spaces MKq(·),p(·)α(·),λ(ℝn) . The boundedness of commutators generated by BMO functions and intrinsic square functions is also discussed on the aforementioned Morrey‐Herz spaces.


INTRODUCTION
For 0 < ≤ 1, let C be the family of all functions ∶ R n → R such that has support contained in {x ∈ R n ∶ |x| ≤ 1}, ∫ R n (x)dx = 0, and such that for any x 1 , x 2 ∈ R the following inequality holds: For ( , t) ∈ R n+1 + and ∈ L 1 loc (R n ), let us set The intrinsic square function of f of order is defined by where t (x) = 1 The definition of intrinsic square function S was first introduced by Wilson. 1,2 Wilson 2 proved the weighed L p boundedness of intrinsic square functions. Lerner 3 proved sharp L p (w) norm inequalities for the intrinsic square function in terms of the A p characteristic of w for all 1 < p < ∞. The boundedness of intrinsic Littlewood-Paley functions on Musielak-Orlicz Morrey and Campanato spaces was considered in Liang et al. 4 Let b ∈ L 1 loc (R n ) such that b ∈ BMO(R n ). The commutator generated by b and the intrinsic square function S (f)(x) is defined by Wang 5 established the commutators of intrinsic square functions [b, S ] on weighted L p space. Guliyev et al. 6 proved the boundedness of intrinsic square function and their commutators on weighted Orlicz-Morrey space.
Moreover, Izuki 7 defined the Herz-Morrey spaces with one variable exponent p(·) and investigated the boundedness of fractional integrals on those space. Lu and Zhu 8 considered the Morrey-Herz spaces M . K (·), q,p(·) (R n ) with two variable exponent (·) and p(·) and obtained some boundedness results for certain sublinear operators and their commutators in these spaces. Wang 9 proved the boundedness of the commutator of the intrinsic square function in variable exponent spaces.
We also mention that Deringoz et al. 10 studied the boundedness of intrinsic square functions and their commutators on vanishing generalized Orlicz-Morrey spaces. Deringoz et al. 10 obtain some conditions for the boundedness are given in terms of Zygmund-type integral inequalities without assuming any monotonicity property.
Finally, it is interesting to point out that the boundedness of several singular integral operator on Herz-type spaces was used in the study of the regularity properties of solutions of second-order elliptic equations with discontinuous coefficients. We mention the work of Ragusa 11 in the context of homogeneous Herz spaces and the works of Scapellato 12,13 in which the authors extended the results contained in Ragusa 11 to Herz spaces with variable exponents. Furthermore, we refer to Ragusa, 14 who studied Herz spaces endowed with a parabolic metric and proved regularity results for weak solutions to divergence form parabolic equations with discontinuous coefficients, using some boundedness results for integral operators and commutators.
The aim of this paper is to discuss boundedness properties of intrinsic square functions and their commutators on the non-homogeneous Morrey-Herz spaces MK (·), q(·),p(·) (R n ) with three variable exponents.

MATHEMATICAL BACKGROUND
Let E be a Lebesgue measurable set in R n with measure |E|>0. Let us denote by E the characteristic function of E. We mention that, throughout the paper, C denotes a positive constant, not necessarily the same at each occurrence. We recall some definitions.
The Lebesgue spaces L p(·) (E) is a Banach spaces with the norm defined by We set p − = ess inf{p(x) ∶ x ∈ E}, p + = ess sup{p(x) ∶ x ∈ E}. (E) is the set of all measurable functions p(·) satisfying p − > 1 and p + < +∞ and  0 (E) is the set of all measurable functions p(·) satisfying p − > 0 and p + < +∞. For any ∈ L 1 loc (R n ), the Hardy-Littlewood maximal operator M is defined by being B a sphere in R n . The set (R n ) consists of all p(·) ∈ (R n ) such that M is bounded on L p(·) (R n ).
If p(·) ∈ (R n ) satisfies the following inequalities, , if | | ≥ |x|, Let us now recall the definition of space BMO(R n ). This space consists of all locally integrable functions f such that where Q = |Q| −1 ∫ Q ( )d , the supremum is taken over all cubes Q ⊂ R n with sides parallel to the coordinate axes, and |Q| denotes the Lebesgue measure of Q. Now, we give the definition of Morrey-Herz space with variable exponents q(·), p(·), (·).
The nonhomogeneous Morrey-Herz space with variable exponents MK (·), q(·),p(·) (R n ) is defined by The homogeneous Morrey-Herz space with variable exponents M If the variable exponents (·), p(·), and q(·) are constants, then M q(·),p(·) (R n ) = L p(·) (R n ). Next, we need some lemmas that will be used in the proofs of our main results.

Lemma 2.3 (Cruz-Uribe & Fiorenza 15 ). (Generalized Hölder's inequality) If p(·)
∈ (R n ), then there exists a constant C such that, for all ∈ L p(·) (R n ) and all g ∈ L p′(·) (R n ), the following inequality holds: Lemma 2.4 (Izuki 17 ). Let p(·) ∈ (R n ). Then, there exists a constant C > 0 such that for any ball B ⊂ R n , the following inequality holds: Lemma 2.5 (Izuki 17 ). Let p(·) ∈ (R n ). For h = 1, 2, there exist constants h1 , h2 , C > 0 such that for all balls B ⊂ R n and all measurable S ⊂ B the following inequalities hold: Lemma 2.7 (Izuki 17 ). Let us assume that b ∈ BMO(R n ) and that n is a positive integer. Then, there exists a constant C > 0, such that for any k, ∈ Z with k > j, the following inequalities hold:

BOUNDEDNESS OF THE INTRINSIC SQUARE FUNCTIONS
Let 1 < p < ∞, p ′ = p p−1 and let w be a weight (i.e., a nonnegative locally integrable function on R n ). We say that w ∈ A p if there exists C > 0 such that for every cube Q ⊆ R n , the following inequality holds: Wilson 1 proved the following weighted (L p − L p ) boundedness of the intrinsic square functions. 19 ). Given a family of functions  , assume that for p 0 , If p(·) ∈ (E), then for all ( , g) ∈  and f ∈ L p(·) (E), we have Since A p/s ′ ⊂ A ∞ , by applying Lemmas 3.1 and 3.2, it is easy to get the boundedness of the intrinsic square functions S on L p(·) . Theorem 3.3. Let us assume that p(·) ∈ (R n ), q 1 (·), q 2 (·) ∈ (R n ) with (q 2 ) − ≥ (q 1 ) + and 0 < ≤ 1. If ( 1 )(q 2 ) + = ( 2 )(q 1 ) − and −n 12 < + < n 11 + ( 1 )/(q 1 ) − , where 11 and 12 are the constants in Lemma 2.5; then, the operator S is bounded from MK + , 1 q 1 (·),p(·) (R n ) to MK (·), 2 q 2 (·),p(·) (R n ). Before starting the proof of Theorem 3.3, we state a simple inequality that will be used in the proof.
. We decompose f as follows:

BMO ESTIMATE FOR THE COMMUTATOR OF INTRINSIC SQUARE FUNCTIONS
Let b ∈ BMO(R n ). Wang 5 obtained some boundedness results for the commutator [b, S ] in the framework of weighted Morrey spaces.
Lemma 4.1. Let 1 < p < ∞, 0 < ≤ 1, and w ∈ A p . Suppose that b ∈ BMO(R n ), then there exists a constant C > 0, independent of f, such that We can apply Lemmas 4.1 and 3.2 to get the boundedness of the commutator [b, S ] in L p(·) .