θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space

Definition 1.1

[7] Let T_θ be a linear operator from 𝓢 into its dual 𝓢^′. One can say that T_θ is a θ-type Calderón-Zygmund operator if it satisfies the following conditions:

1. T_θ can be extended to be a bounded linear operator on L²(ℝⁿ) ;

2. there is a θ-type Calderón-Zygmund kernel K(x, y) such that

   Tθf(x):=∫RnK(x,y)f(y)dy,  as  f∈Cc∞(Rn)  and  x∉suppf.(1.4)

   It is easy to see that the classical Calderón-Zygmund operator with standard kernel is a special case of θ-type operator T_θ as θ(t) = t^δ with 0 < δ ≤ 1. Given a locally integrable function b, the commutator generated by T_θ and b is defined by

   [b,Tθ]f(x)=b(x)Tθf(x)−Tθ(b⋅f)(x)=∫Rn[b(x)−b(y)]K(x,y)f(y)dy.(1.5)

Such type of operators are extensively applied in PDE with non-smooth area. Many authors concentrate on the boundedness of these operators on various function spaces, we refer the reader to see [8, 9, 10, 11, 12, 13, 14] for its developments and applications. In [11], Quek-Yang established the boundedness of T_θ on spaces such as weighted Lebesgue spaces, weighted weak Lebesgue spaces, weighted Hardy spaces and weighted weak Hardy spaces. Ri-Zhang obtained the bounedness of T_θ on Hardy spaces with non-doubling measures and non-homogeneous metric measure spaces in [12, 13]. Wang proved the boundedness of T_θ and [b, T_θ] on the generalized weighted Morrey spaces in [14]. Inspired by the results mentioned previously, a natural and interesting problem is to consider whether or not the θ-type Calderón-Zygmund operators T_θ and their commutators [b, T_θ] are bounded on Herz space with variable exponents. The purpose of this paper is to give a positive answer.

The spaces with variable exponent have been widely studied in recent ten years. The results show that they are not only the generalized forms of the classical function spaces with invariable exponent, but also there are some new breakthroughs in the research techniques. These new real variable methods help people further understand the function spaces. Lebesgue spaces with variable exponent L^p(⋅)(ℝⁿ) become one class of important function spaces due to the fundamental paper [15] by Kovóčik Rákosník. The theory of the function spaces with variable exponent have been applied in fluid dynamics, elastlcity dynamics, calculus of variations and differential equations with non-standard growth conditions(for example, see [16, 17, 18, 19, 20]). In [21], authors proved the extrapolation theorem which leads the boundedness of some classical operators including the commutators on L^p(⋅)(ℝⁿ). Karlovich and Lerner also obtained the bundedness of the singular integral commutators in [22]. The boundedness of some typical operators is being studied with keen interest on spaces with variable exponent (see [23, 24, 25, 26]). Recently, Tao and Yang established the boundedness of θ-Type C-Z operators and their commutators generated respectively by BMO functions, Lipschitz functions and Besov functions on variable exponent Lebesgue spaces L^p(⋅)(ℝⁿ) (see [27]), at the same time, the boundedness of singular integrals with variable kernel and fractional differentiations is also obtained.

It is well known that Herz spaces play an important role in harmonic analysis. After the Herz spaces with one exponent p(⋅) were introduced in [28], the boundedness of some operators and some characterizations of these spaces were studied widely (see [29, 30, 31, 32]). In this paper, we will study the boundedness of the θ-type Calderón-Zygmund operators T_θ and their commutators [b, T_θ] on Herz spaces with two variable exponents p(⋅), q(⋅). In order to do this, we need to recall some notations and definitions.

Denote 𝓟(ℝⁿ) to be the set of the all measurable functions p(x) with p₋ =: ess infx∈Rnp(x) > 1 and p₊ =: ess supx∈Rnp(x) < ∞ and 𝓑(ℝⁿ) to be the set of all functions p(⋅) ∈ 𝓟(ℝⁿ) satisfying the condition that the Hardy-Littlewood maximal operator M is bounded on L^p(⋅)(ℝⁿ), 𝓟⁰(ℝⁿ) the set of all measurable functions p(x) with p₋ > 0 and p₊ < ∞.

Given a function p(x) ∈ 𝓟(ℝⁿ), the space L^p(x)(ℝⁿ) is now defined by^[15]

Given a function p(⋅) ∈ 𝓟⁰(ℝⁿ), the space L^p(⋅)(ℝⁿ) is defined by

It is easy to see that the above Luxemburg-Nakano quasi-norm is also equivalent with^[21]

where q(x)=p(x)p0 and for some p₀ with 0 < p₀ < p₋.

Before recalling the Herz spaces with two variable exponents, we will introduce the following function space, which is named as the mixed Lebesgue sequence space (see [33]).

Definition 1.2

^{[24, 33]} Let p(⋅), q(⋅) ∈ 𝓟⁰(ℝⁿ). The mixed Lebesgue sequence space with variable exponent l^q(⋅)(L^p(⋅)) is the collection of all sequences {fj}j=0∞ of measurable functions on ℝⁿ such that

where

Noting q₊ < ∞, we see that

Let k ∈ ℤ, B_k = {x ∈ ℝⁿ : |x| ≤ 2^k}, C_k = B_k∖B_k−1, χ_k = χ_Ck.

Definition 1.3

^[24] Let α ∈ ℝ, p(⋅), q(⋅) ∈ 𝓟⁰(ℝⁿ). The homogeneous Herz space with variable exponent K˙p(⋅)α,q(⋅)(Rn) is defined by

where

Remark 1.1

1. It is easy to see that K˙p(⋅)0,p(⋅)(Rn) = L^p(⋅)(ℝⁿ) and if p(x) = p₀ and q(x) = q₀ are constants, then K˙p(⋅)α,q(⋅)(Rn)=K˙p0α,q0(Rn) is just the usual Herz spaces (see [34]).

2. If q₁(⋅), q₂(⋅) ∈ 𝓟⁰(ℝⁿ) satisfying (q₁)₊ ≤ (q₂)₋, then (see [24])

   K˙p(⋅)α,q1(⋅)(Rn)⊂K˙p(⋅)α,q2(⋅)(Rn).

Definition 1.5

^[35] When 0 < β ≤ 1, the Homogeneous Lipschitz space Lip_β(ℝⁿ) is the space of functions such that

Our main results in this paper are fomulated as follows.

Theorem 1.1

Suppose that T_θ is a θ-type Calderón-Zygmund operators with θ satisfies (1, 1), p(⋅) ∈ 𝓑(ℝⁿ), q₁(⋅), q₂(⋅) ∈ 𝓟⁰(ℝⁿ) with (q₂)₋ ≥ (q₁)₊ and p(⋅)/q₂(⋅) ∈ 𝓟(ℝⁿ). If −nδ₁₂ < α < nδ₁₁, where δ₁₁, δ₁₂ are the constants in Lemma 2.2, then the operator T_θ is bounded from K˙p(⋅)α,q1(⋅)(Rn) to K˙p(⋅)α,q2(⋅)(Rn).

Theorem 1.2

Let b ∈ BMO, m ∈ ℕ and p(⋅), q₁(⋅), q₂(⋅) and α are the same as in Theorem 1.1. Suppose that [b, T_θ] is defined by (1.5) with θ satisfying

Then the commutator [b, T_θ] is bounded from K˙p(⋅)α,q1(⋅)(Rn) to K˙p(⋅)α,q2(⋅)(Rn).

Theorem 1.3

Let b ∈ Lip_β(ℝⁿ)(0 < β < 1) and [b, T_θ] be defined by (1.5) with θ satisfying (1.6). Suppose that q₁(⋅), q₂(⋅) ∈ 𝓟⁰(ℝⁿ) with (q₂)₋ ≥ (q₁)₊ and p₁(⋅), p₂(⋅) ∈ 𝓑(ℝⁿ) is such that (p₁)₊ < nβ, 1/p₁(x)−1/p₂(x) = β/n, (1−β/n)p₂(⋅) ∈ 𝓑(ℝⁿ) and p₂(⋅)/q₂(⋅) ∈ 𝓟(ℝⁿ). If −nδ₁₂ < α < nδ₁₁, then the commutator [b, T_θ] is bounded from K˙p1(⋅)α,q1(⋅)(Rn) to K˙p2(⋅)α,q2(⋅)(Rn).

We make some conventions. In what follows, C always denotes a positive constant which is independent of the main parameters involved but whose value may differ in different occurrences, |E| denotes the Lebesgue measure of E ∈ ℝⁿ. Given a function f, we denote the mean value of f on E by f_E =: 1|E|∫_Ef(x)dx. p^′(⋅) means the conjugate exponent of p(⋅), namely, 1/p(x) + 1/p^′(x) = 1 holds.

Lemma 2.1

^[15] (Generalized Hölder’s Inequality) Let p(⋅), p₁(⋅), p₂(⋅) ∈ p(⋅) ∈ 𝓟(ℝⁿ).

1. For any f ∈ L^p(⋅)(ℝⁿ) and g ∈ L^p′(⋅)(ℝⁿ),

   ∫Rn|f(x)g(x)|dx≤Cp∥f∥Lp(⋅)∥g∥Lp′(⋅),

   where C_p = 1 + 1/p⁻ − 1/p⁺.

2. For any f ∈ L^p₁(⋅)(ℝⁿ) and g ∈ Lp2′(⋅)(Rn), when 1/p(x) = 1/p₁(x) + 1/p₂(x), we have

   ∥f(x)g(x)∥Lp(⋅)(Rn)≤Cp1,p2∥f∥Lp1(⋅)∥g∥Lp2′(⋅),

   where C_p1, p₂ = (1 + 1/p₁₋ − 1/p_1⁺)^1/p₋.

Lemma 2.2

^[22] If p_i ∈ 𝓑(ℝⁿ) (i = 1, 2), then there exist constants p_{i, 1}, p_{i, 2}, C > 0, such that for all balls B ∈ ℝⁿ and all measurable subsets S ∈ B,

Lemma 2.3

^[30] If p_i(⋅) ∈ 𝓑(ℝⁿ), then there exists a constant C > 0 such that for all balls B ∈ ℝⁿ,

Lemma 2.4

^[24] Let p(⋅), q₁(⋅) ∈ 𝓟(ℝⁿ). If f ∈ L^p(⋅)q(⋅)(ℝⁿ), then

Lemma 2.5

^[30] Let b ∈ BMO and m be a positive integer. There exists a constant C > 0, such that for any k, j ∈ ℤ with k > j,

1. C⁻¹∥b∥∗m≤supB1∥χB∥Lp(⋅)(Rn) ∥(b − b_B)^mχ_B∥_{L^p(⋅)(ℝⁿ)} ≤ C∥b∥∗m;

2. ∥(b − b_Bj)^mχ_Bk∥_{L^p(⋅)(ℝⁿ)} ≤ C(k − j)^m∥b∥∗m ∥χ_Bk∥_{L^p(⋅)(ℝⁿ)}.

Lemma 2.6

^[36] If p(⋅) ∈ 𝓑(ℝⁿ), then there exist constants 0 < δ < 1, C > 0, such that for all Y ∈ 𝓣, all nonnegative numbers t_Q and all f ∈ Lloc1(ℝⁿ) with f_Q ≠ 0 (Q ∈ Y),

where 𝓣 denotes all families of disjoint and open cube in ℝⁿ.

Lemma 2.7

^[24] Let b ∈ Lip_β(ℝⁿ) and m be a positive integer. There exists a constant C > 0, such that for any k, j ∈ ℤ with k > j

1. C⁻¹∥b∥Lipβm≤1|B|mβ/n∥χB∥Lp(⋅)(Rn)−1∥(b − b_B)^mχ_B∥_{L^p(⋅)(ℝⁿ)} ≤ C∥b∥Lipβm;

2. ∥(b − b_Bj)^mχ_Bk ∥_{L^p(⋅)(ℝⁿ)} ≤ C|B_k|^mβ/n∥b∥Lipβm ∥χ_B∥_{L^p(⋅)(ℝⁿ)}.

Lemma 2.8

^[37] Let f ∈ Lip_β(ℝⁿ), 0 < β < 1, 1 ≤ p < ∞, B₁ ⊂ B₂. We have

1. ∥f∥_Lipβ ≈ sup_B1|B|β/n(1|B|∫B|f(x)−fB|pdx)1/p;

2. |f_B1 − f_B2| ≤ C∥f∥_Lipβ|B₂|^β/n.

Lemma 2.9

^[24] Let k ∈ ℕ, a_k ≥ 0, 1 ≤ p_k < ∞. Then

Lemma 2.10

^[27] Suppose that T_θ is a θ-type Calderón-Zygmund operator and θ satisfies (1, 1). Let p(⋅) ∈ 𝓑(ℝⁿ), then there exits a constant C independent of f such that

Lemma 2.11

^[27] Let b ∈ BMO(ℝⁿ). Suppose that p(⋅) ∈ 𝓑(ℝⁿ) and θ satisfies (1.6), then there exists a constant C independent of f such that

Lemma 2.12

^[27] Let b ∈ Lip_β(ℝⁿ) for 0 < β < 1. Suppose that θ satisfies (1, 1) and p(⋅) ∈ 𝓟(ℝⁿ) be such that p₊ < nβ. Define q(⋅) by

If q(⋅)(1 − βn) ∈ 𝓑(ℝⁿ), then there exists a constant C independent of f such that

Proof of Theorem 1.1

Let f ∈ K˙p(⋅)α,q1(⋅)(Rn). Write

Due to p(⋅)/q₂(⋅) ∈ 𝓟(ℝⁿ), which implies inf(p(⋅)/q₂(⋅)) ≥ 1, then we have

Since

where

Hence,

This implies that, in order to prove our theorem, we only need to show η₁₁, η₁₂, η₁₃ ≤ C∥f∥K˙p(⋅)α,q1(⋅)(Rn). For simplicity, we denote λ₀ = ∥f∥K˙p(⋅)α,q1(⋅)(Rn).

First, we consider η₁₂. By Lemma 2.4 and using Aoki-Rolewicz’s theorem, we have

where

By using Lemma 2.10, it follows

Since f ∈ K˙p(⋅)α,q1(⋅)(Rn), we can easily see 2kα|fχk|λ0Lp(⋅) ≤ 1 and ∑k=−∞∞2kα|fχk|λ0q1(⋅)Lp(⋅)q1(⋅) ≤ 1. Hence, by Lemma 2.4 and Lemma 2.9, we have

Here (q₁)₊ ≤ (q₂)₋ ≤ (q21)k and q∗=mink∈Z(q21)k(q1)+. Consequently, we have η₁₂ ≤ Cλ₀.

Now let us turn to estimate η₁₁.

Let x ∈ C_k, y ∈ C_j, j ≤ k − 3, then we have |x − y| ≥ |x| − |y| ≥ 38|x|. Hence, we have

Thus, By Lemmas 2.1-2.4 and the fact that 2jα|fχj|λ0q1(⋅)Lp(⋅)q1(⋅) ≤ 1, it follows that

where

If (q₁)₊ < 1, then by Lemma 2.9 and the fact (q₁)₊ ≤ (q₂)₋ ≤ (q21)k, we have

where q∗=mink∈Z(q22)k(q1)+.

If (q₁)₊ ≥ 1, then 1 ≤ (q₁)₊ ≤ (q₂)₋ ≤ (q21)k. Thus, for α < nδ₁₁, by applying Hölder’s inequality and Lemma 2.9, we get