Singular integrals with variable kernel and fractional di erentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents

Let T be the singular integral operator with variable kernel de ned by Tf(x) = p.v.∫ Rn Ω(x, x − y) ∣x − y∣n f(y)dy and D(0 ≤ γ ≤ 1) be the fractional di erentiation operator. Let T and T be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TD −DT and (T−T♯)D on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents HMK̇ p(⋅),λ via the convolution operator Tm,j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on HMK̇ p(⋅),λ(R n) is shown to hold for TD −DT and (T −T♯)D . Moreover, the authors also establish various norm characterizations for the product T1T2 and the pseudo-product T1 ○ T2.

Boundedness properties of the above operator in a variety of functional spaces have been extensively studied. In particular, Calderón and Zygmund proved that T is bounded on the L (R n ) in the Mihlin conditions (see [1]). Other references with results of this sort include [2][3][4][5] and the references within. On the other hand, these estimates played an important role in the theory of non-divergent elliptic equations with discontinuous coe cients (see [6,7]).
We make some conventions. In what follows, χ E denotes the characteristic function for a µ-measurable set E. We use the symbol A ≲ B to denote that there exsists a positive constant C such that A ≤ CB. For any index p ∈ ( , ∞), we denote by p ′ its conjugate index, that is, p + p ′ = .
Denote T * and T ♯ to be the adjoint of T and the pseudo-adjoint of T respectively (see (3.2) and (3.3) below). Let T and T be the operators de ned in (1.1) which are di erentiateded by its kernel Ω (x, y) and Ω (x, y). Let T T , T ○T denote the product and pseudo-product of T and T , respectively. In 1957, Calderón and Zygmund found that these operators are closely related to the second order linear elliptic equations with variable coe cients and established the following results of the operators T * , T ♯ , T T , T ○ T and D on L p (R n )( < p < ∞) (see [1]).
Theorem A ( [1]). Let < p < ∞, Ω (x, y), Ω (x, y) ∈ C β (C ∞ ), β > satisfy (1.1) and (1.2). Then there is a constant C such that (1) In 2015, Chen and Zhu proved that Theorem A was also true on Weighted Lebesgue space and Morrey space (see [9]). In 2016, Tao and Yang obtained the boundedness of those operators on the weighted Morrey-Herz spaces (see [5]), Later, the boundedness of those operators on the Lebesgue spaces with variable exponents were obtained [10]. Inspired by the ideas mentioned previously, the aim of this paper is to deal with the boundedness of the singular integrals with variable kernel and fractional di erentiations in the setting of the Morrey-Herz-type Hardy Spaces with variable exponents (which will be de ned in the next section).
The main theorems are presented in this section. The de nitions of the Morrey-Herz spaces with variable exponents, the Morrey-Herz-type Hardy spaces with variable exponents and the preliminary lemmas are presented in Section 2. In Section 3, we will introduce the spherical harmonical expansions and give the boundedness of T m,j . The proofs of Theorems are given in Section 4.

De nitions and preliminaries
In this section, the Morrey-Herz spaces with variable exponents MK α(⋅),q p(⋅),λ and the Morrey-Herz-type Hardy spaces with variable exponents HMK α(⋅),q p(⋅),λ will be introduced. Some preliminary lemmas will be given as well.
Lebesgue spaces with variable exponent L p(⋅) (R n ) become one of important function spaces due to the fundamental paper [11] by Kovócik Rákosník. In the past 20 years, the theory of these spaces have made progress rapidly. On the other hand, the function spaces with variable exponent have been applied in uid dynamics, elastlcity dynamics, calculus of variations and di erential equations with non-standard growth conditions (see [12][13][14][15][16]). In [17], authors proved the extrapolation theorem which leads to the boundedness of some classical operators including the commutators on L p(⋅) (R n ). Karlovich and Lerner also obtained the bundedness of the singular integral commutators in [18]. The boundedness of some typical operators has been studied with keen interest (see [18][19][20][21][22][23][24][25]). Recently, Xu and Yang have introduced the Morrey-Herz-type Hardy spaces with variable exponents and established the boundedness of singular integral operators on these spaces in [26].
De nition 2.1 ([21]). Let α(⋅) be a real-valued function on R n . If there exist C > such that for any x, y ∈ R n , x − y < , , then α(⋅) is said to be local log-Hölder continuous on R n . If there exist C > such that for all x ∈ R n , , then α(⋅) is said to be log-Hölder continuous at origin. If there exist α ∞ ∈ R and a constant C > such that all x ∈ R n , , then α(⋅) is said to be log-Hölder continuous at in nity.
Equipped with the Luxemburg-Nakano norm Then P(R n ) consists of all p(⋅) satisfying p − > and p + < ∞.

Let M be te Hardy-littlewood maximal operator. We denote B(R n ) to be the set of all functions p(⋅) ∈ P(R n ) satisfying the condition that M is bounded on L p(⋅) (R n ).
We now make some conventions. Throughout this paper, let k ∈ Z,

De nition 2.3 ([26]
). Let < q ≤ ∞, p(⋅) ∈ P(R n ), and ≤ λ < ∞. Let α(⋅) be a bounded real-valued measurable function on R n . The homogeneous Morrey-Herz space MK with the corresponding modi cation for q = ∞. Next let us recall the de nition of Morrey-Herz-type Hardy spaces with variable exponents HMK α(⋅),q p(⋅),λ , which was rstly introduced by Xu and Yang in [26]. To do this, we need some natations. S(R n ) denotes the Schwartz spaces of all rapidly decreasing in nitely di erentiable functions on R n , and S ′ (R n ) denotes the dual space of S(R n ). Let G N f be the grand maximal function of f de ned by for any x ∈ R n and t > . The grand maximal G N was rstly introduced by Fe erman and Stein in [27] to study classical Hardy spaces. We refer the reader to [28][29][30] for details on the classical Hardy spaces. The variable exponent case is shown in [22] by Nakai and Sawano.
De nition 2.5 ([26]). Let p(⋅) ∈ P(R n ) and α(⋅) ∈ L ∞ (R n ) be log-Hölder continuous both at the origin and in nity, and nonnegative integer s where the in mum is taken over all above decompositions of f .

Spherical harmonics and boundedness of T m,l
In this section, we will recall the spherical harmonical expansion and give the boundedness of T m,l , which are very vital in our proofs of Theorems.
We let H m denote the space of spherical harmonics homogeneous polynomials of degree m. Let dimH m = d m and {Y m,j } d m j= be an orthonormal system of H m . It is showed that {Y m,j } d m j= , m = , , . . . , is a complete orthonormal system in L (S n− ) (see [32]). We can expand the kernel Ω(x, z ′ ) in spherical harmonics as Then T, de ned in (1.3), can be written as Let T * and T ♯ be the adjoint of T and the pseudo-adjoint of T respectively, de ned by and T m,j f MK α(⋅),λ For simplicity, we denote Φ = ∑ L∈Z −Lλ ∑ To complete our proof, we only need to show that I, II, III ≲ m nq Φ. First, we estimate I: By the result that T m,l is bounded on L p(⋅) (R n ) (see [10]), we have Therefore, when we get < q ≤ , we get As < q < ∞, we can obtain Hence, we have I ≲ m nq Φ. Secondly, we estimate I . A simple computation shows that there exists a constant δ > such that T m,l satis es the following size condition and with the help of Lemma 2.8, we get ≲ m n j(δ−α j )−k(δ+n) χ B j L p ′ (⋅) . So by Lemma 2.6 and 2.8, we have Therefore, when < q ≤ , by nδ ≤ α( ) < δ + nδ we get When < q < ∞, let q + q ′ = . Since nδ ≤ α( ) ≤ δ + nδ , by Hölder's inequality, we have Hence, we have I ≲ m nq Φ. Thirdly, we estimate II. Consider When < q ≤ , we get As < q < ∞, we obtain For II , as < q ≤ , noting that nδ ≤ α( ) < δ + nδ , we have As < q < ∞ and nδ ≤ α( ) < δ + nδ , by Hölder's inequality, one has So, we have II ≲ m nq Φ. Finally, we estimate III. Write When < q ≤ , by the boundedness of T m,l in L p(⋅) (R n ) (see [10]), we obtain As < q < ∞, by the boundedness of T m,l in L p(⋅) (see [10]) and Hölder's inequality, we have For III , as < q ≤ and nδ ≤ α( ), α ∞ < δ + nδ , we get As < q < ∞ and nδ ≤ α( ), α ∞ ≤ δ + nδ , by Hölder's inequality, we have Thus, we have III ≲ m nq Φ.
Combining the estimates I, II, III, we complete the proof of Lemma 3.1.