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

Theorem A

([1]). Let 1 < p < ∞, Ω₁(x, y), Ω₂(x, y) ∈ C^β(C^∞), β > 1 satisfy (1.1) and (1.2). Then there is a constant C such that

1. ∥(T₁D − DT₁)f∥_L^p ≲ ∥f∥_L^p;

2. ∥(T1∗−T1♯)Df∥Lp≲∥f∥Lp;

3. ∥(T₁ ∘ T₂ − T₁T₂)Df ∥_L^p ≲ ∥f∥_L^p.

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 differentiations in the setting of the Morrey-Herz-type Hardy Spaces with variable exponents (which will be defined in the next section).

The main theorems are presented in this section. The definitions 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.

Theorem 1.1

Let p(⋅) ∈ 𝓑(ℝⁿ), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), nδ₂ ≤ α(0), α_∞ ≤ nδ₂+δwith some δ > max{α(0) − δ₂, α_∞ − nδ₂} and δ₂as in Lemma 2.6. Assume that T is defined by (1.3) and Ω(x, y), which satisfies (1.1), (1.2), meet a condition

then we have

1. ∥(TDγ−DγT)f∥MK˙p(⋅),λα(⋅),q≲∥f∥HMK˙p(⋅),λα(⋅),q;

2. ∥(T∗−T♯)Dγf∥MK˙p(⋅),λα(⋅),q≲∥f∥HMK˙p(⋅),λα(⋅),q.

Theorem 1.2

Let p(⋅) ∈ 𝓑(ℝⁿ), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), nδ₂ ≤ α(0), α_∞ ≤ nδ₂+δ with some δ > max{α(0) − δ₂, α_∞ − nδ₂} and δ₂as in Lemma 2.6. Suppose that Ω₁(x, y) and Ω₂(x, y) satisfy (1.1) and (1.2). If Ω₂(x, y) satisfies (1.4) and Ω₁(x, y) satisfies

then we have

Theorem 1.3

Let p(⋅) ∈ 𝓑(ℝⁿ), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), nδ₂ ≤ α(0), α_∞ ≤ nδ₂+δ with some δ > max{α(0) − δ₂, α_∞ − nδ₂} and δ₂as in Lemma 2.6. Suppose that Ω_i(x, y)(i = 1, 2) satisfies (1.1), (1.2) and (1.5), then we have

1. ∥(T₁𝓘 − 𝓘T₁)f∥MK˙p(⋅),λα(⋅),q≲∥f∥HMK˙p(⋅),λα(⋅),q;

2. ∥(T1∗−T2♯)If∥MK˙p(⋅),λα(⋅),q≲∥f∥HMK˙p(⋅),λα(⋅),q;

3. ∥(T₁ ∘ T₂ − T₁T₂)𝓘f∥MK˙p(⋅),λα(⋅),q≲∥f∥HMK˙p(⋅),λα(⋅),q;

Theorem 1.4

Let p(⋅) ∈ 𝓑(ℝⁿ), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2λ ≤ α(⋅), nδ₂ ≤ α(0), α_∞ ≤ nδ₂+δ with some δ > max{α(0) − δ₂, α_∞ − nδ₂} and δ₂as in Lemma 2.6. Suppose that Ω(x, y) satisfies(1.1), (1.2) and

then we have

1. ∥(TD − DT)f∥MK˙p(⋅),λα(⋅),q≲∥f∥HMK˙p(⋅),λα(⋅),q;

2. ∥(T^∗ − T^♯)Df∥MK˙p(⋅),λα(⋅),q≲∥f∥HMK˙p(⋅),λα(⋅),q.

Theorem 1.5

Let p(⋅) ∈ 𝓑(ℝⁿ), 0 < q < ∞, and 0 ≤ λ < ∞. If α(⋅) is a bounded and log-Hölder continuous both at the origin and infinity such that 2 λ ≤ α(⋅), nδ₂ ≤ α(0), α_∞ ≤ nδ₂+δ with some δ > max{α(0) − δ₂, α_∞ − nδ₂} and δ₂as in Lemma 2.6. Suppose that Ω₁(x, y) and Ω₂(x, y) satisfies(1.1), (1.2). If Ω₁(x, y) satisfies (1.5) and Ω₂(x, y) satisfies (1.6), then we have

Definition 2.1

([21]). Let α(⋅) be a real-valued function on ℝⁿ. If there exist C > 0 such that for any x, y ∈ ℝⁿ, |x − y| < 1/2,

then α(⋅) is said to be local log-Hölder continuous on ℝⁿ.

If there exist C > 0 such that for all x ∈ ℝⁿ,

then α(⋅) is said to be log-Hölder continuous at origin.

If there exist α_∞ ∈ ℝ and a constant C > 0 such that all x ∈ ℝⁿ,

then α(⋅) is said to be log-Hölder continuous at infinity.

Definition 2.2

([11]). Let p: ℝⁿ ⟶ [1, ∞) be a measurable function. The Lebesgue space with variable exponent L^p(⋅)(ℝⁿ) is defined by

Equipped with the Luxemburg-Nakano norm

we denote

Then 𝓟(ℝⁿ) consists of all p(⋅) satisfying p₋ > 1 and p₊ < ∞.

Let M be te Hardy-littlewood maximal operator. We denote 𝓑(ℝⁿ) to be the set of all functions p(⋅) ∈ 𝓟(ℝⁿ) satisfying the condition that M is bounded on L^p(⋅)(ℝⁿ).

We now make some conventions. Throughout this paper, let k ∈ ℤ, B_k = {x ∈ ℝⁿ : |x| ≤ 2^k}, C_k = B_k∖ B_{k − 1}, L ∈ ℤ and χ_k = χ_Ck.

Definition 2.3

([26]). Let 0 < q ≤ ∞, p(⋅) ∈ 𝓟(ℝⁿ), and 0 ≤ λ < ∞. Let α(⋅) be a bounded real-valued measurable function on ℝⁿ. The homogeneous Morrey-Herz spaceMK˙p(⋅),λα(⋅),qis defined by

where

with the corresponding modification for q = ∞.

Next let us recall the definition of Morrey-Herz-type Hardy spaces with variable exponents HMK˙p(⋅),λα(⋅),q, which was firstly introduced by Xu and Yang in [26]. To do this, we need some natations. 𝓢(ℝⁿ) denotes the Schwartz spaces of all rapidly decreasing infinitely differentiable functions on ℝⁿ, and 𝓢^′(ℝⁿ) denotes the dual space of 𝓢(ℝⁿ). Let G_Nf be the grand maximal function of f defined by

where AN={ϕ∈S(Rn):sup|ν|,|β|≤N,∀x∈Rn|xνDβϕ(x)|≤1} and N > n+1 and ϕ∇∗ is the nontangential maximal operator defined by ϕ∇∗(f)(x)=sup|y−x|<t|ϕt∗f(y)|, where ϕt(x)=t−nϕ(xt) for any x ∈ ℝⁿ and t > 0.

The grand maximal G_N was firstly introduced by Fefferman 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.

Definition 2.4

([26]). Let α(⋅) ∈ L^∞(ℝⁿ), 0 < q ≤ ∞, p(⋅) ∈ 𝓟(ℝⁿ), 0 ≤ λ < ∞, and N > n+1. The homogeneous Morrey-Herz-type Hardy space with variable exponentsHMK˙p(⋅),λα(⋅),qis defined by

Obviously, if α(⋅) = α and λ = 0, these spaces were considered by Wang and Liu in [23]. If p(⋅) and α(⋅) are constant and λ = 0, these are the classical Herz type Hardy spaces (see [31]).

Definition 2.5

([26]). Let p(⋅) ∈ 𝓟(ℝⁿ) and α(⋅) ∈ L^∞(ℝⁿ) be log-Hölder continuous both at the origin and infinity, and nonnegative integer s ≥ [α_r − nδ₂], here α_r = α(0), if r < 1, and α_r = α_∞, if r ≥ 1, nδ₂ ≤ α_r < ∞ and δ₂as in Lemma 2.6.

1. A function a on ℝⁿis called a central (α(⋅), p(⋅))-atom, if it satisfies (1) supp a ⊂ B(0, r); (2) ∥a∥_L^p(⋅) ≤ |B(0, r)|^−α_r/n; (3) ∫_ℝⁿa(x)x^βdx = 0, |β| ≤ s.

2. A function a on ℝⁿis called a central (α(⋅), p(⋅))-atom of restricted type, if it satisfies (1) supp a ⊂ B(0, r), r ≥ 1; (2) ∥a∥_L^p(⋅) ≤ |B(0, r)|^−α_r/n; (3) ∫_ℝⁿa(x)x^βdx = 0, |β| ≤ s.

Lemma 2.6

([18]). If p(⋅) ∈ 𝓑(ℝⁿ), then there exist constants δ₁, δ₂ > 0, such that for all balls B ⊂ ℝⁿand all measurable subsets S ⊂ B,

Lemma 2.7

([21, Theorem 13]). Let 0 < q < ∞, p(⋅) ∈ 𝓑(ℝⁿ), 0 ≤ λ < ∞, and α(⋅) ∈ L^∞be log-Hölder continuous both at the origin and infinity, 2λ ≤ α(⋅), nδ₂ ≤ α(0), α_∞ < ∞ with δ₂as in Lemma 2.5, then f ∈ HMK˙p(⋅),λα(⋅),qif and only iff=∑k=−∞∞λkakin the sense of 𝓢^′(ℝⁿ), where each a_k is a central (α(⋅), p(⋅))-atom with support contained in B_k andsupL∈Z2−Lλ∑k=−∞L|λk|q<∞.Moreover,

where the infimum is taken over all above decompositions of f.

Lemma 2.8

([11]). Let p(⋅) ∈ 𝓟(ℝⁿ). If f ∈ L^p(⋅)and g ∈ L^p′(⋅), then fg is integrable on ℝⁿand

where r_p = 1+1/p⁻ − 1/p⁺.

Lemma 2.9

([26]). Let α(⋅) be a bounded and log-Hölder continuous both at the origin and infinity such that nδ₂ ≤ α(0), α_∞ < nδ₂+δ with some δ > max{α(0) − nδ₂, α_∞ − nδ₂} and δ₂as in Lemma 2.6. Suppose T is a Calderón-Zygmund operator associated to a standerd kernel K. If p(⋅) ∈ 𝓑(ℝⁿ), 0 < λ < ∞ and 0 ≤ q < ∞, then we have

Lemma 2.10

([26]). Let p(⋅) ∈ 𝓟(ℝⁿ), q ∈ (0, ∞], and λ ∈ (0, ∞]. Ifα(⋅)∈L∞(Rn)∩P0log(Rn)∩P∞log(Rn),then

Lemma 3.1

Let α(⋅) ∈ L^∞be log-Hölder continuous both at the origin and infinity, 2λ ≤ α(⋅), nδ₂ ≤ α(0), α_∞ < nδ₂+δ with some δ > max{α(0) − nδ₂, α_∞ − nδ₂} and δ₂as in Lemma 2.6. The T_{m, j}defined by (3.1) is bounded fromHMK˙p(⋅),qα(⋅),λtoMK˙p(⋅),qα(⋅),λ,i.e.

Proof

Suppose f ∈ MK˙p(⋅),λα(⋅),q. By Lemma 2.7, f=∑j=−∞∞λjbj converges in 𝓢^′(ℝⁿ), where each b_j is a central (α(⋅), q(⋅)) – atom with support contained in B_j and

For simplicity, we denote Φ=∑L∈Z2−Lλ∑j=−∞L|λj|q. By Lemma 2.10, we have

where

To complete our proof, we only need to show that I, II, III ≲ m^nq/2Φ. First, we estimate I:

By the result that T_{m, l} is bounded on L^p(⋅)(ℝⁿ) (see [10]), we have

Therefore, when we get 0 < q ≤ 1, we get

As 1 < q < ∞, we can obtain