Boundedness of Commutators of High-Dimensional Hausdorff Operators

Abstract

We prove the boundedness of commutators of high-dimensional Hausdorff operators 𝐻_Φ,𝑏 on Lebesgue space with central BMO function or Lipschitz function. Furthermore, the boundedness on Herz spaces and Morrey-Herz spaces is also obtained.

1. Introduction

The one-dimensional Hausdorff operator is defined asℎ_Φ𝑓(𝑥)=^∞0Φ(𝑡)𝑡𝑓𝑥𝑡𝑑𝑡,(1.1) where Φ is a locally integrable function on (0,∞). The operator ℎ_Φ has a deep root in the study of the one-dimensional Fourier analysis. Particularly, it is closely related to the summability of the classical Fourier series. The reader can see ([1–3]) to find details of background and recent development of the Hausdorff operator.

Recently, Hausdorff operator in the high dimensional space ℝ^𝑛 was studied. Three extensions of the one-dimensional Hausdorff operator in ℝ^𝑛 were recently introduced and studied in [4]. One of them is the operator𝐻_Φ𝑓(𝑥)=_ℝ𝑛Φ||𝑦||⁻¹𝑥||𝑦||^𝑛𝑓(𝑦)𝑑𝑦,(1.2) where Φ is a radial function defined on ℝ⁺. Replacing Φ withΦ₁(𝑡)=𝑡^−𝑛𝜒_(1,∞)(𝑡),Φ₂(𝑡)=𝜒_(0,1)(𝑡),(1.3)𝐻_Φ𝑓 becomes the high dimensional Hardy operator𝐻𝑓(𝑥)=1|𝑥|^𝑛_{|𝑦|<|𝑥|}𝑓(𝑦)𝑑𝑦,𝑥∈ℝ^𝑛⧵{0},(1.4) and its adjoint operator𝐻^∗𝑓(𝑥)=_{|𝑦|≥|𝑥|}𝑓(𝑦)||𝑦||^𝑛𝑑𝑦,(1.5) respectively. It is well known that Hardy operators are important operators in Harmonic analysis and a quite number of papers have appeared in [5–9]. And in [10], the authors have obtained the boundedness of commutators of one-dimensional Hausdorff operator with one-sided dyadic CMO functions on the Lebesgue space.

These observations motivate us to study the boundedness of commutators of 𝑛-dimensional Hausdorff operators on some function spaces. Let 𝑏(𝑥) be a real measurable, locally integrable function; we define the commutators of Hausdorff operators as follows:𝐻_Φ,𝑏𝑓(𝑥)=𝑏(𝑥)𝐻_Φ𝑓(𝑥)−𝐻_Φ(𝑏𝑓)(𝑥)=_ℝ𝑛Φ||𝑦||⁻¹𝑥||𝑦||^𝑛=(𝑏(𝑥)−𝑏(𝑦))𝑓(𝑦)𝑑𝑦_{|𝑦|<|𝑥|}Φ||𝑦||⁻¹𝑥||𝑦||^𝑛+(𝑏(𝑥)−𝑏(𝑦))𝑓(𝑦)𝑑𝑦_{|𝑦|≥|𝑥|}Φ||𝑦||⁻¹𝑥||𝑦||^𝑛∶=𝐻_1Φ,𝑏(𝑏(𝑥)−𝑏(𝑦))𝑓(𝑦)𝑑𝑦𝑓(𝑥)+𝐻^2Φ,𝑏𝑓(𝑥).(1.6)

Definition 1.1 (see [11]). Let 1<𝑞<∞. A function 𝑓∈𝐿^𝑞loc(𝑅^𝑛) is said to belong to the space CṀO^𝑞(ℝ^𝑛) if ‖𝑓‖_Ċ‌MO^𝑞_(ℝ^𝑛₎=sup_𝑟>0||𝐵(10,𝑟)||_𝐵(0,𝑟)||𝑓(𝑥)−𝑓_𝐵||^𝑞𝑑𝑥^1/𝑞<∞,(1.7) where 𝑓_𝐵=(1/|𝐵(0,𝑟)|)∫_𝐵(0,𝑟)𝑓(𝑥)𝑑𝑥. Obviously, we have BMO(ℝ^𝑛)⊂CṀO^𝑞(ℝ^𝑛) for all 1≤𝑞<∞, and CṀO^𝑞(ℝ^𝑛)⊂CṀO^𝑝(ℝ^𝑛), 1≤𝑝<𝑞<∞.

Definition 1.2 (see [12]). Let 0<𝛾<1. The Besov-Lipschitz space Λ̇_𝛾(ℝ^𝑛) is defined by ‖𝑏‖_̇Λ𝛾(ℝ^𝑛₎=sup_{𝑥,ℎ∈ℝ𝑛}||𝑏(𝑥+ℎ)−𝑏(𝑥)||||ℎ||^𝛾<∞.(1.8)

Herz-type spaces are important function spaces in harmonic analysis. It should be pointed out that Lu and Yang make tremendous contributions on this spaces. Their book (joint with Hu) [13] is the unique research book in this topic. Below, we briefly recall the definition of the Herz-type spaces. We denote by 𝐵(𝑥,𝑟) the ball centered at 𝑥 with radius 𝑟. 𝐶 is a constant which may vary from line to line. For 𝑘∈ℤ, let 𝐵_𝑘={𝑥∈ℝ^𝑛∶|𝑥|≤2^𝑘}, Δ_𝑘=𝐵_𝑘⧵𝐵_𝑘−1, and 𝜒_𝑘(𝑘∈ℤ) denote the characteristic function of the set Δ_𝑘.

Definition 1.3 (see [13]). Let 𝛼∈ℝ,0<𝑝<∞, and 0<𝑞<∞. The homogeneous Herz space 𝐾̇^{𝑞𝛼,𝑝}(ℝ^𝑛) is defined by 𝐾̇^{𝑞𝛼,𝑝}(ℝ^𝑛)=𝑓∈𝐿^𝑞loc(ℝ^𝑛⧵{0})∶‖𝑓‖_̇𝐾^{𝑞𝛼,𝑝}_(ℝ^𝑛₎<∞,(1.9) where ‖𝑓‖_̇𝐾^{𝑞𝛼,𝑝}_(ℝ^𝑛₎=^∞_𝑘_=−∞2^𝑘𝛼𝑝‖‖𝑓𝜒_𝑘‖‖^𝑝𝐿_{𝑞(ℝ𝑛)}^1/𝑝.(1.10)

Obviously, 𝐾̇^𝑞0,𝑞(ℝ^𝑛)=𝐿^𝑞(ℝ^𝑛) and 𝐾̇^{𝑞𝛼/𝑞,𝑞}(ℝ^𝑛)=𝐿^𝑞(ℝ^𝑛,|𝑥|^𝛼), so the Herz space is the natural generalization of the Lebesgue spaces with power weight |𝑥|^𝛼.

Definition 1.4 (see [13]). Let 𝛼∈ℝ,0<𝑝≤∞,0<𝑞<∞ and 𝜆≥0. The homogeneous Morrey-Herz space 𝑀𝐾̇^{𝛼,𝜆𝑝,𝑞}(ℝ^𝑛) is defined by 𝑀𝐾̇^{𝛼,𝜆𝑝,𝑞}(ℝ^𝑛)=𝑓∈𝐿^𝑞loc(ℝ^𝑛⧵{0})∶‖𝑓‖_𝑀̇𝐾^{𝛼,𝜆𝑝,𝑞}_(ℝ𝑛)<∞,(1.11) where ‖𝑓‖_𝑀̇𝐾^{𝛼,𝜆𝑝,𝑞}_(ℝ𝑛)=sup_𝑘0∈ℤ2^−𝑘0𝜆⎛⎜⎜⎝^𝑘0_𝑘_=−∞2^𝑘𝛼𝑝‖‖𝑓𝜒_𝑘‖‖^𝑝𝐿_{𝑞(ℝ𝑛)}⎞⎟⎟⎠^1/𝑝(1.12) with the usual modification made when 𝑝=∞.

In [14], the Morrey space 𝑀^𝜆𝑞(ℝ^𝑛) is defined by𝑀^𝜆𝑞(ℝ^𝑛)=𝑓∈𝐿^𝑞loc(ℝ^𝑛)∶sup_{𝜆>0,𝑥∈ℝ𝑛}1𝑟^𝜆_{|𝑥−𝑦|<𝑟}||𝑓(𝑦)||^𝑞𝑑𝑦<∞.(1.13) Obviously, 𝑀𝐾̇^{𝛼,0𝑝,𝑞}(ℝ^𝑛)=𝐾̇^{𝑞𝛼,𝑝}(ℝ^𝑛) and 𝑀^𝜆𝑞(ℝ^𝑛)⊂𝑀𝐾̇^{0,𝜆𝑞,𝑞}(ℝ^𝑛).

2. Main Results

Now, we state our main results.

Theorem 2.1. Let 1<𝑝<∞, 1<𝑟<min(𝑝,𝑝^), and 𝑏∈CṀO^{max(𝑝,𝑝𝑟/(𝑝−𝑟))}(ℝ^𝑛):(a)if one sets 𝐴_Φ,𝑟=^∞0||Φ(𝑡)||^𝑟𝑡^{−1+𝑛(𝑟/𝑟)}𝑑𝑡^1/𝑟<∞,(2.1) then 𝐻^1Φ,𝑏 is bounded on 𝐿^𝑝(ℝ^𝑛);(b)if one sets 𝐵_Φ,𝑟=^∞0||Φ(𝑡)||^𝑟𝑡^−1+𝑛𝑑𝑡^1/𝑟<∞,(2.2) then 𝐻^2Φ,𝑏 is bounded on 𝐿^𝑝(ℝ^𝑛);(c)if Φ satisfies (2.1) and (2.2), then 𝐻_Φ,𝑏 is bounded on 𝐿^𝑝(ℝ^𝑛).

Remark 2.2. When Φ(𝑡)=𝑡^−𝑛𝜒_(1,∞)(𝑡), 𝐻_Φ,𝑏𝑓(𝑥)=𝐻^1Φ,𝑏𝑓(𝑥)=𝐻_𝑏𝑓(𝑥). We can check that Φ(𝑡) satisfies (2.1); therefore, the boundedness of commutator of Hardy operator is obtained. When Φ(𝑡)=𝜒_(0,1)(𝑡), 𝐻_Φ,𝑏𝑓(𝑥)=𝐻^2Φ,𝑏𝑓(𝑥)=𝐻^∗𝑏𝑓(𝑥). We also know that Φ(𝑡) satisfies (2.2); therefore, we get the boundedness of commutator of the adjoint Hardy operator see [9].

Remark 2.3. Using the same method, we can get the generalized result of Theorem 2.1(b). Let 1<𝑝<∞,1<𝑟<𝑝, 0≤𝜆<min(𝑟′/𝑟,𝑟′/𝑝), and 𝑏∈CMO^{max(𝑝,𝑝𝑟/(𝑝−𝑟))}(ℝ^𝑛). If 𝐵_Φ,𝑟,𝜆=^∞0||Φ(𝑡)||^𝑟𝑡^{−1+𝑛𝜆}𝑑𝑡^1/𝑟<∞,(2.3) then 𝐻^2Φ,𝑏 is bounded on 𝐿^𝑝(ℝ^𝑛).

Theorem 2.4. Let 1<𝑝,𝑞<∞,1<𝑟<min(𝑝,𝑞′),1/𝑝−1/𝑞=𝛾/𝑛, and 𝑏∈Λ̇_𝛾(ℝ^𝑛)(0<𝛾<1). Then(a)if Φ satisfies (2.1), then 𝐻^1Φ,𝑏 are bounded from 𝐿^𝑝(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛);(b)if Φ satisfies (2.2), then 𝐻^2Φ,𝑏 are bounded from 𝐿^𝑝(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛);(c)if Φ satisfies (2.1) and (2.2), then 𝐻_Φ,𝑏 is bounded from 𝐿^𝑝(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛).

Remark 2.5. Just like in Remark 2.2, we get Lipschitz estimates for commutator of Hardy operator or the adjoint Hardy operator. See the details in [15].

Theorem 2.6. Let 0<𝑝<∞,1<𝑞<∞,1<𝑟<𝑞, and 𝑏∈CṀO^{max(𝑞,𝑞𝑟/(𝑞−𝑟))}(ℝ^𝑛). Then(a)if Φ satisfies (2.1) and 𝛼<𝑛(1/𝑟−1/𝑞), then 𝐻^1Φ,𝑏 is bounded on 𝐾̇^{𝑞𝛼,𝑝}(ℝ^𝑛);(b)if Φ satisfies (2.2) and 𝛼>−𝑛(1/𝑞−1/𝑟′), then 𝐻^2Φ,𝑏 is bounded on 𝐾̇^{𝑞𝛼,𝑝}(ℝ^𝑛);(c)if Φ satisfies (2.1), (2.2), and −𝑛(1/𝑞−1/𝑟′)<𝛼<𝑛(1/𝑟−1/𝑞), then 𝐻_Φ,𝑏 is bounded on 𝐾̇^{𝑞𝛼,𝑝}(ℝ^𝑛).

Remark 2.7. Let 𝛼=0 in Theorem 2.6, then Theorem 2.1 can be obtained.

Theorem 2.8. Let 0<𝑝₁≤𝑝₂<∞, 1<𝑞₁, 𝑞₂<∞, 1<𝑟<𝑞₁, 1/𝑞₁−1/𝑞₂=𝛾/𝑛, and 𝑏∈Λ̇_𝛾(ℝ^𝑛)(0<𝛾<1). Then(a)if Φ satisfies (2.1) and 𝛼<𝑛(1/𝑟−1/𝑞₁), then 𝐻^1Φ,𝑏 is bounded from 𝐾̇^𝛼,𝑝1_𝑞1(ℝ^𝑛) to 𝐾̇^𝛼,𝑝2_𝑞2(ℝ^𝑛);(b)if Φ satisfies (2.2) and 𝛼>−𝑛(1/𝑞₂−1/𝑟′), then 𝐻^2Φ,𝑏 are bounded from 𝐾̇^𝛼,𝑝1_𝑞1(ℝ^𝑛) to 𝐾̇^𝛼,𝑝2_𝑞2(ℝ^𝑛);(c)if Φ satisfies (2.1), (2.2), and −𝑛(1/𝑞₂−1/𝑟′)<𝛼<𝑛(1/𝑟−1/𝑞₁), then 𝐻_Φ,𝑏 is bounded from 𝐾̇^𝛼,𝑝1_𝑞1(ℝ^𝑛) to 𝐾̇^𝛼,𝑝2_𝑞2(ℝ^𝑛).

Remark 2.9. Let 𝛼=0 in Theorem 2.8, then Theorem 2.4 can be obtained.

Theorem 2.10. Let 0<𝑝<∞, 1<𝑞<∞, 1<𝑟<𝑞,𝜆≥0, and 𝑏∈CṀO^{max(𝑞,𝑞𝑟/(𝑞−𝑟))}(ℝ^𝑛). Then(a)if Φ satisfies (2.1) and 𝛼<𝑛(1/𝑟−1/𝑞)+𝜆, then 𝐻^1Φ,𝑏 is bounded on 𝑀𝐾̇^{𝛼,𝜆𝑝,𝑞}(ℝ^𝑛);(b)if Φ satisfies (2.2) and 𝛼>−𝑛(1/𝑞−1/𝑟′)+𝜆, then 𝐻^2Φ,𝑏 is bounded on 𝑀𝐾̇^{𝛼,𝜆𝑝,𝑞}(ℝ^𝑛);(c)if Φ satisfies (2.1), (2.2), and −𝑛(1/𝑞−1/𝑟′)+𝜆<𝛼<𝑛(1/𝑟−1/𝑞)+𝜆, then 𝐻_Φ,𝑏 is bounded on 𝑀𝐾̇^{𝛼,𝜆𝑝,𝑞}(ℝ^𝑛).

Theorem 2.11. Let 𝜆≥0,0<𝑝₁≤𝑝₂<∞,1<𝑞₁,𝑞₂<∞,1<𝑟<𝑞₁,1/𝑞₁−1/𝑞₂=𝛾/𝑛, and 𝑏∈Λ̇_𝛾(ℝ^𝑛)(0<𝛾<1).(a)if Φ satisfies (2.1) and 𝛼<𝑛(1/𝑟−1/𝑞₁)+𝜆, then 𝐻^1Φ,𝑏 is bounded from 𝑀𝐾̇^{𝑝𝛼,𝜆}_1,𝑞1(ℝ^𝑛) to 𝑀𝐾̇^{𝑝𝛼,𝜆}_2,𝑞2(ℝ^𝑛);(b)if Φ satisfies (2.2) and 𝛼>−𝑛(1/𝑞₂−1/𝑟′)+𝜆, then 𝐻^2Φ,𝑏 is bounded from 𝑀𝐾̇^{𝑝𝛼,𝜆}_1,𝑞1(ℝ^𝑛) to 𝑀𝐾̇^{𝑝𝛼,𝜆}_2,𝑞2(ℝ^𝑛);(c)if Φ satisfies (2.1), (2.2), and −𝑛(1/𝑞₂−1/𝑟′)+𝜆<𝛼<𝑛(1/𝑟−1/𝑞₁)+𝜆, then 𝐻_Φ,𝑏 is bounded from 𝑀𝐾̇^{𝑝𝛼,𝜆}_1,𝑞1(ℝ^𝑛) to 𝑀𝐾̇^{𝑝𝛼,𝜆}_2,𝑞2(ℝ^𝑛).

3. Proof of the Main Results

We first give several lemmas.

Lemma 3.1. Let 1<𝑟≤𝑝<∞ and 𝜆≥0, then(i)∫_Δ𝑖(|Φ(𝑥/|𝑦|)|/|𝑦|^𝑛)|𝑓(𝑦)|𝑑𝑦≤𝐶𝐴_Φ,𝑟2^{𝑖𝑛(1/𝑟−1/𝑝)}|𝑥|^−𝑛/𝑟‖𝑓𝜒_𝑖‖_{𝐿𝑝(ℝ𝑛)}, (ii)∫_Δ𝑖(|Φ(𝑥/|𝑦|)|/|𝑦|^𝑛)|𝑓(𝑦)|𝑑𝑦≤𝐶𝐵_Φ,𝑟,𝜆2^{𝑖𝑛(𝜆/𝑟′−1/𝑝)}|𝑥|^{−𝑛𝜆/𝑟′}‖𝑓𝜒_𝑖‖_{𝐿𝑝(ℝ𝑛)}.

Proof. (i)By Hölder’s inequality and polar coordinate, we obtain _Δ𝑖||Φ𝑥/||𝑦||||||𝑦||^𝑛||𝑓(𝑦)||𝑑𝑦≤_Δ𝑖||Φ𝑥/||𝑦||||^𝑟||𝑦||^𝑛𝑟𝑑𝑦^1/𝑟′_Δ𝑖||𝑓(𝑦)||^𝑟𝑑𝑦^1/𝑟≤^{|𝑥|/2𝑖−1}_|𝑥|/2𝑖||𝑆^𝑛−1||||Φ(𝑡)||^𝑟𝑡^{−1+𝑛𝑟/𝑟}|𝑥|^{𝑛(1−𝑟)}𝑑𝑡^1/𝑟×_Δ𝑖||𝑓(𝑦)||^𝑝𝑑𝑦^1/𝑝2^{𝑖𝑛(1/𝑟−1/𝑝)}≤𝐶𝐴_Φ,𝑟2^{𝑖𝑛(1/𝑟−1/𝑝)}|𝑥|^−𝑛/𝑟‖‖𝑓𝜒_𝑖‖‖_{𝐿𝑝(ℝ𝑛)}.(3.1)(ii)By Hölder’s inequality and polar coordinate, we have _Δ𝑖||Φ𝑥/||𝑦||||||𝑦||^𝑛||𝑓(𝑦)||𝑑𝑦≤_Δ𝑖||Φ(𝑥/|𝑦|)||^𝑟||𝑦||^{(𝑛(𝜆+1)/𝑟)𝑟}𝑑𝑦^1/𝑟_Δ𝑖||𝑓(𝑦)||^𝑟||𝑦||^{(𝑛−𝑛(𝜆+1)/𝑟)𝑟}𝑑𝑦^1/𝑟≤𝐶^2𝑖_2𝑖−1_𝑆𝑛−1||Φ|𝑥|𝑅⁻¹||^𝑟𝑅^{−1−𝑛𝜆}𝑑𝜎𝑦^𝑑𝑅^1/𝑟×2^{−𝑖𝑛(1−(𝜆+1)/𝑟)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑟≤𝐶^2𝑖_2𝑖−1_𝑆𝑛−1||Φ|𝑥|𝑅⁻¹||^𝑟𝑅^{−1−𝑛𝜆}𝑑𝜎𝑦^𝑑𝑅^1/𝑟×2^{𝑖𝑛(𝜆/𝑟−1/𝑝)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑝≤𝐶𝐵_Φ,𝑟,𝜆2^{𝑖𝑛(𝜆/𝑟−1/𝑝)}|𝑥|^{−𝑛𝜆/𝑟}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑝.(3.2)

Lemma 3.2 (see [9]). Let 𝑏 be a CṀO¹(ℝ^𝑛) function and 𝑖, 𝑘∈ℤ, then ||𝑏(𝑡)−𝑏_𝐵𝑖||≤||𝑏(𝑡)−𝑏_𝐵𝑘||+2||𝑖−𝑘||‖𝑏‖_{Ċ‌MO1(ℝ𝑛)}.(3.3)

Lemma 3.3 (see [12]). Let 0<𝛾<1 and 𝑓∈Λ̇_𝛾(ℝ^𝑛). Then ||𝑓(𝑥)−𝑓(𝑦)||≤||𝑥−𝑦||^𝛾‖𝑓‖_̇Λ𝛾(ℝ^𝑛₎≤|𝑥|^𝛾+||𝑦||^𝛾‖𝑓‖^̇Λ_𝛾(ℝ^𝑛₎.(3.4)

Proof of Theorem 2.6. (a) When Φ satisfies (2.1) and 𝛼<𝑛(1/𝑟−1/𝑞), we get ‖‖𝐻^1Φ,𝑏𝑓𝜒_𝑘‖‖^𝑞𝐿_𝑞≤𝐶_𝑞_Δ𝑘^𝑘_𝑖=−∞_Δ𝑖||Φ𝑥/||𝑦||||||𝑦||^𝑛+𝐶_𝑞||𝑏(𝑥)−𝑏_𝐵𝑘𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥_Δ𝑘^𝑘_𝑖=−∞_Δ𝑖||Φ𝑥/||𝑦||||||𝑦||^𝑛∶=𝐼+𝐼𝐼.||𝑏(𝑦)−𝑏_𝐵𝑘𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥(3.5) For 𝐼, using Lemma 3.1(i), we have 𝐼≤𝐶_𝑞𝐴^𝑞Φ,𝑟_𝐵𝑘||𝑏(𝑥)−𝑏_𝐵𝑘||^𝑞^𝑘_𝑖=−∞2^{𝑖𝑛(1/𝑟−1/𝑞)}|𝑥|^−𝑛/𝑟‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞≤𝐶_𝑞𝑑𝑥𝐴^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO𝑞(ℝ𝑛)}||𝐵_𝑘||^𝑘_𝑖=_−∞2^{𝑖𝑛(1/𝑟−1/𝑞)}2^{−𝑘𝑛/𝑟}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞=𝐶_𝑞𝐴^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO𝑞(ℝ𝑛)}^𝑘_𝑖=−∞2^{(𝑖−𝑘)𝑛(1/𝑟−1/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞.(3.6) For 𝐼𝐼, by Lemma 3.2, we have 𝐼𝐼≤𝐶_𝑞_Δ𝑘^𝑘_𝑖=−∞_Δ𝑖||Φ𝑥/||𝑦||||||𝑦||^𝑛+𝐶_𝑞||𝑏(𝑦)−𝑏_𝐵𝑖𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥_Δ𝑘^𝑘_𝑖=−∞_Δ𝑖||Φ𝑥/||𝑦||||||𝑦||^𝑛∶=𝐼𝐼₁(𝑘−𝑖)‖𝑏‖_Ċ‌MO¹_(ℝ^𝑛₎||𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥+𝐼𝐼₂.(3.7) For 𝐼𝐼₁, by Lemma 3.1(i) and Hölder’s inequality, we have 𝐼𝐼₁≤𝐶_𝑞𝐴^𝑞Φ,𝑟_Δ𝑘^𝑘_𝑖=−∞|𝑥|^−𝑛/𝑟_Δ𝑖||𝑏(𝑦)−𝑏_𝐵𝑖𝑓(𝑦)||^𝑟𝑑𝑦^1/𝑟^𝑞≤𝐶_𝑞𝑑𝑥𝐴^𝑞Φ,𝑟_Δ𝑘^𝑘_𝑖=−∞|𝑥|^−𝑛/𝑟_Δ𝑖||𝑏(𝑦)−𝑏_𝐵𝑖||^{𝑟𝑞/(𝑞−𝑟)}𝑑𝑦^{(𝑞−𝑟)/𝑞𝑟}_Δ𝑖||𝑓(𝑦)||^𝑞𝑑𝑦^1/𝑞^𝑞≤𝐶_𝑞𝐴^𝑞Φ,𝑟^𝑘_𝑖=−∞2^{𝑘𝑛(1/𝑞−1/𝑟)}‖𝑏‖_Ċ‌MO^{𝑞𝑟/(𝑞−𝑟)}_(ℝ^𝑛₎||𝐵_𝑖||^{(𝑞−𝑟)/𝑞𝑟}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞=𝐶_𝑞𝐴^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO𝑞𝑟/(𝑞−𝑟)(ℝ𝑛)}^𝑘_𝑖=−∞2^{(𝑖−𝑘)𝑛(1/)−1/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞.(3.8) For 𝐼𝐼₂, by Lemma 3.1(i), we have 𝐼𝐼₂≤𝐶_𝑞𝐴^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO1(ℝ𝑛)}_Δ𝑘^𝑘_𝑖=−∞(𝑘−𝑖)2^{𝑖𝑛(1/𝑟−1/𝑞)}|𝑥|^−𝑛/𝑟‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞≤𝐶_𝑞𝑑𝑥𝐴^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO1(ℝ𝑛)}2^𝑘𝑛^𝑘_𝑖=−∞(𝑘−𝑖)2^{𝑖𝑛(1/𝑟−1/𝑞)}2^{−𝑘𝑛/𝑟}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞=𝐶_𝑞𝐴^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO1(ℝ𝑛)}^𝑘_𝑖=−∞(𝑘−𝑖)2^{(𝑖−𝑘)𝑛(1/𝑟−1/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞.(3.9)
Following the estimates of 𝐼,𝐼𝐼₁, and 𝐼𝐼₂, we can obtain that ‖‖𝐻^1Φ,𝑏𝑓‖‖_{̇𝐾𝑞𝛼,𝑝(ℝ𝑛)}≤𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^𝑞_(ℝ^𝑛₎⎛⎜⎜⎝^∞_𝑘_=−∞2^𝑘𝛼𝑝^𝑘_𝑖=−∞2^{(𝑖−𝑘)𝑛(1/𝑟−1/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑝⎞⎟⎟⎠^1/𝑝+𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^{𝑞𝑟/(𝑞−𝑟)}_(ℝ^𝑛₎⎛⎜⎜⎝^∞_𝑘_=−∞2^𝑘𝛼𝑝^𝑘_𝑖=−∞2^{(𝑖−𝑘)𝑛(1/𝑟−1/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑝⎞⎟⎟⎠^1/𝑝+𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO¹_(ℝ^𝑛₎⎛⎜⎜⎝^∞_𝑘_=−∞2^𝑘𝛼𝑝^𝑘_𝑖=−∞(𝑘−𝑖)2^{(𝑖−𝑘)𝑛(1/𝑟−1/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑝⎞⎟⎟⎠^1/𝑝∶=𝑆.(3.10) Obviously, 𝑆≤𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^{max(𝑞,𝑞𝑟/(𝑞−𝑟))}_(ℝ^𝑛₎⎛⎜⎜⎝^∞_𝑘_=−∞2^𝑘𝛼𝑝^𝑘_𝑖=−∞(𝑘−𝑖)2^{(𝑖−𝑘)𝑛(1/𝑟−1/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑝⎞⎟⎟⎠^1/𝑝.(3.11) When 0<𝑝≤1, we get 𝑆^𝑝≤𝐶_𝑝𝐴^𝑝Φ,𝑟‖𝑏‖^𝑝_{Ċ‌MOmax(𝑞,𝑞𝑟/(𝑞−𝑟))(ℝ𝑛)∞𝑘}_𝑘=−∞_𝑖=−∞2^𝑖𝛼𝑝‖‖𝑓𝜒_𝑖‖‖^𝑝𝐿_𝑞(𝑘−𝑖)^𝑝2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞−𝛼)𝑝}=𝐶_𝑝𝐴^𝑝Φ,𝑟‖𝑏‖^𝑝_{Ċ‌MOmax(𝑞,𝑞𝑟/(𝑞−𝑟))(ℝ𝑛)∞𝑖=}_−∞2^𝑖𝛼𝑝‖‖𝑓𝜒_𝑖‖‖^𝑝𝐿_𝑞^∞_𝑘=𝑖(𝑘−𝑖)^𝑝2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞−𝛼)𝑝}=𝐶_𝑝𝐴^𝑝Φ,𝑟‖𝑏‖^𝑝_{Ċ‌MOmax(𝑞,𝑞𝑟/(𝑞−𝑟))(ℝ𝑛)}‖𝑓‖^𝑝̇𝐾_{𝑞𝛼,𝑝}.(3.12) For the case 𝑝>1, it follows from Hölder’s inequality that 𝑆^𝑝≤𝐶_𝑝𝐴^𝑝Φ,𝑟‖𝑏‖^𝑝_{Ċ‌MOmax(𝑞,𝑞𝑟/(𝑞−𝑟))(ℝ𝑛)∞𝑘}_𝑘=−∞_𝑖=−∞2^𝑖𝛼𝑝‖‖𝑓𝜒_𝑖‖‖^𝑝𝐿_𝑞2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞−𝛼)𝑝/2}×^𝑘_𝑖=_−∞(𝑘−𝑖)^𝑝2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞−𝛼)𝑝/2}^𝑝/𝑝≤𝐶_𝑝𝐴^𝑝Φ,𝑟‖𝑏‖^𝑝_{Ċ‌MOmax(𝑞,𝑞𝑟/(𝑞−𝑟))(ℝ𝑛)∞𝑖=}_−∞2^𝑖𝛼𝑝‖‖𝑓𝜒_𝑖‖‖^𝑝𝐿_𝑞^∞_𝑘=𝑖2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞−𝛼)𝑝/2}=𝐶_𝑝𝐴^𝑝Φ,𝑟‖𝑏‖^𝑝_{Ċ‌MOmax(𝑞,𝑞𝑟/(𝑞−𝑟))(ℝ𝑛)}‖𝑓‖^𝑝̇𝐾_{𝑞𝛼,𝑝}.(3.13) When Φ satisfies (2.2) and 𝛼>−𝑛(1/𝑞−1/𝑟′), we get ‖‖(𝐻^2Φ,𝑏𝑓)𝜒_𝑘‖‖^𝑞𝐿_{𝑞(ℝ𝑛)}≤𝐶_𝑞_Δ𝑘^∞_𝑖=𝑘_Δ𝑖|Φ(𝑥/|𝑦|)||𝑦|^𝑛+𝐶_𝑞||𝑏(𝑥)−𝑏_𝐵𝑘𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥_𝐶𝑘^∞_𝑖=𝑘_Δ𝑖|Φ(𝑥/|𝑦|)||𝑦|^𝑛∶=𝐽+𝐽𝐽.||𝑏(𝑦)−𝑏_𝐵𝑘𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥(3.14) For 𝐽, using Lemma 3.1(ii), we have 𝐽≤𝐶_𝑞𝐵^𝑞Φ,𝑟_Δ𝑘||𝑏(𝑥)−𝑏_𝐵𝑘||^𝑞^∞_𝑖=𝑘2^{𝑖𝑛(1/𝑟′−1/𝑞)}|𝑥|^{−𝑛/𝑟′}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞≤𝐶_𝑞𝑑𝑥𝐵^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO𝑞(ℝ𝑛)}||𝐵_𝑘||^∞_𝑖=𝑘2^{𝑖𝑛(1/𝑟′−1/𝑞)}2^{−𝑘𝑛/𝑟}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞=𝐶_𝑞𝐵^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO𝑞(ℝ𝑛)}^∞_𝑖=𝑘2^{(𝑘−𝑖)𝑛(1/𝑞−1/𝑟′)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞.(3.15) For 𝐽𝐽, by Lemma 3.2, we have 𝐽𝐽≤𝐶_𝑞_Δ𝑘^∞_𝑖=𝑘_Δ𝑖||Φ(𝑥/|𝑦|)||||𝑦||^𝑛+𝐶_𝑞||𝑏(𝑦)−𝑏_𝐵𝑖𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥_Δ𝑘^∞_𝑖=𝑘_Δ𝑖||Φ(𝑥/|𝑦|)||||𝑦||^𝑛∶=𝐽𝐽₁(𝑖−𝑘)‖𝑏‖_Ċ‌MO¹_(ℝ^𝑛₎||𝑓(𝑦)||𝑑𝑦^𝑞𝑑𝑥+𝐽𝐽₂.(3.16) For 𝐽𝐽₁, by Lemma 3.1(ii) and Hölder’s inequality, similarly to 𝐼𝐼₁, we have 𝐽𝐽₁≤𝐶_𝑞𝐵^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MOqr/(q-r)(ℝ𝑛)}^∞_𝑖=𝑘2^{(𝑘−𝑖)𝑛(1/𝑞−1/𝑟′)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞.(3.17) For 𝐽𝐽₂, also due to Lemma 3.1(ii), we have 𝐽𝐽₂≤𝐶_𝑞𝐵^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO1(ℝ𝑛)}_Δ𝑘^∞_𝑖=𝑘(𝑖−𝑘)2^{𝑖𝑛(1/𝑟−1/𝑞)}|𝑥|^{−𝑛/𝑟′}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞≤𝐶_𝑞𝑑𝑥𝐵^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO1(ℝ𝑛)}2^𝑘𝑛^∞_𝑖=𝑘(𝑖−𝑘)2^{𝑖𝑛(1/𝑟−1/𝑞)}2^{−𝑘𝑛/𝑟′}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞=𝐶_𝑞𝐵^𝑞Φ,𝑟‖𝑏‖^𝑞_{Ċ‌MO1(ℝ𝑛)}^∞_𝑖=𝑘(𝑖−𝑘)2^{(𝑘−𝑖)𝑛(1/𝑞−1/𝑟)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑞.(3.18) The remaining proof is similar to the proof of (a), so that of (b) can be obtained easily.
(c) When Φ satisfies (2.1), (2.2), and −𝑛(1/𝑞−1/𝑟′)<𝛼<𝑛(1/𝑟−1/𝑞), due to 𝐻_Φ,𝑏𝑓(𝑥)=𝐻^1Φ,𝑏𝑓(𝑥)+𝐻^2Φ,𝑏𝑓(𝑥),(3.19) therefore, by (a) and (b), we can easily get (c).

Proof of Theorem 2.8. (a) When Φ satisfies (2.1) and 𝛼<𝑛(1/𝑟−1/𝑞₁), by Lemmas 3.3, 3.1(i), and 1/𝑞₁−1/𝑞₂=𝛾/𝑛, we get ‖‖𝐻^1Φ,𝑏𝑓𝜒_𝑘‖‖^𝑞2_𝐿𝑞2≤‖𝑏‖^𝑞2_̇Λ𝛾_Δ𝑘^𝑘_𝑖=−∞_Δ𝑖||Φ(𝑥/|𝑦|)|||𝑦|^𝑛≤𝐶‖𝑏‖_{𝑞2̇Λ𝛾}|𝑥|^𝛾+||𝑦||^𝛾||𝑓(𝑦)||𝑑𝑦^𝑞2𝑑𝑥_Δ𝑘|𝑥|^𝛾𝑘_𝑖=−∞_Δ𝑖||Φ(𝑥/|𝑦|)|||𝑦|^𝑛≤𝐶_𝑞2|𝑓(𝑦)|𝑑𝑦^𝑞2𝑑𝑥𝐴^𝑞2_Φ,𝑟‖𝑏‖^𝑞2_̇Λ𝛾2^𝑘𝑛2^{𝑘𝑘𝛾−𝑘𝑛/𝑟}_𝑖=−∞2^{𝑖𝑛(1/𝑟−1/𝑞1)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞1^𝑞2=𝐶_𝑞2𝐴^𝑞2_Φ,𝑟‖𝑏‖^𝑞2_̇Λ𝛾^𝑘_𝑖=−∞2^{(𝑖−𝑘)𝑛(1/𝑟−1/𝑞1)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞1^𝑞2.(3.20) The remaining proof Is similar to the proof of (a) of Theorem 2.6, so that of (a) can be obtained easily.
(b) When Φ satisfies (2.2) and 𝛼>−𝑛(1/𝑞₂−1/𝑟′), by Lemmas 3.3, 3.1(ii), and (1/𝑞₁)−(1/𝑞₂)=𝛾/𝑛, we get ‖‖(𝐻^2Φ,𝑏𝑓)𝜒_𝑘‖‖^𝑞2_{𝐿𝑞2(ℝ𝑛)}≤‖𝑏‖^𝑞2_̇Λ𝛾_Δ𝑘^∞_𝑖=𝑘_Δ𝑖||Φ(𝑥/|𝑦|)|||𝑦|^𝑛≤𝐶_𝑞2|𝑥|^𝛾+||𝑦||^𝛾||𝑓(𝑦)||𝑑𝑦^𝑞2𝑑𝑥‖𝑏‖^𝑞2_̇Λ𝛾_Δ𝑘^∞_𝑖=𝑘_Δ𝑖|𝑦|^𝛾||Φ(𝑥/|𝑦|)|||𝑦|^𝑛≤𝐶_𝑞2|𝑓(𝑦)|𝑑𝑦^𝑞2𝑑𝑥𝐵^𝑞2_Φ,𝑟‖𝑏‖^𝑞2_̇Λ𝛾_Δ𝑘^∞_𝑖=𝑘2^𝑖𝛾2^{𝑖𝑛(1/𝑟′−1/𝑞1)}|𝑥|^{−𝑛/𝑟′}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞1^𝑞2≤𝐶_𝑞2𝑑𝑥𝐵^𝑞2_Φ,𝑟‖𝑏‖^𝑞2_̇Λ𝛾^∞_𝑖=𝑘2^{(𝑘−𝑖)𝑛(1/𝑞2−1/𝑟′)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞1^𝑞2.(3.21) The remaining proof is similar to the proof of (a) of Theorem 2.6, so that of (b) can be obtained easily.
(c) The proof is the same as the proof of (c) of Theorem 2.6.

Proof of Theorem 2.10. We only give the proof of (a) when 𝜆>0. The proof of (b) is similar to that for (a), and the proof of (c) is similar to that for (c) of Theorem 2.6. By the definition of Morrey-Herz spaces and the estimates for 𝐼, 𝐼𝐼₁, and 𝐼𝐼₂ above, we have ‖‖𝐻^1Φ,𝑏𝑓‖‖_{𝑀̇𝐾𝛼,𝜆𝑝,𝑞}≤𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^𝑞_(ℝ^𝑛₎sup_𝑘0∈ℤ2^−𝑘0𝜆⎛⎜⎜⎝^𝑘0_𝑘_=−∞2^𝑘𝛼𝑝^𝑘_𝑖=−∞2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑝⎞⎟⎟⎠^1/𝑝+𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^{𝑞𝑟/(𝑞−𝑟)}_(ℝ^𝑛₎sup_𝑘0∈ℤ2^−𝑘0𝜆⎛⎜⎜⎝^𝑘0_𝑘_=−∞2^𝑘𝛼𝑝^𝑘_𝑖=−∞2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑝⎞⎟⎟⎠^1/𝑝+𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO¹_(ℝ^𝑛₎sup_𝑘0∈ℤ2^−𝑘0𝜆⎛⎜⎜⎝^𝑘0_𝑘_=−∞2^𝑘𝛼𝑝^𝑘_𝑖=−∞(𝑘−𝑖)2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞)}‖‖𝑓𝜒_𝑖‖‖_𝐿𝑞^𝑝⎞⎟⎟⎠^1/𝑝∶=𝐸₁+𝐸₂+𝐸₃.(3.22) For 𝐸₁, noting that 𝛼<𝑛(1/𝑟−1/𝑞)+𝜆, we have 𝐸₁≤𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^𝑞sup_𝑘0∈ℤ2^−𝑘0𝜆⎛⎜⎜⎝^𝑘0_𝑘_=−∞⎛⎜⎜⎝^𝑘_𝑖=_−∞2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞−𝛼)}^𝑖_𝑙=−∞2^𝑙𝛼𝑝‖‖𝑓𝜒_𝑖‖‖^𝑝𝐿_𝑞^1/𝑝⎞⎟⎟⎠^𝑝⎞⎟⎟⎠^1/𝑝≤𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^𝑞sup_𝑘0∈ℤ2^−𝑘0𝜆⎛⎜⎜⎝^𝑘0_𝑘_=−∞2^𝑘𝑝𝜆^𝑘_𝑖=−∞2^{(𝑖−𝑘)(𝑛/𝑟−𝑛/𝑞−𝛼+𝜆)}‖𝑓‖_𝑀̇𝐾^{𝛼,𝜆𝑝,𝑞}^𝑝⎞⎟⎟⎠^1/𝑝≤𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^𝑞sup_𝑘0∈ℤ2^−𝑘0𝜆⎛⎜⎜⎝^𝑘0_𝑘_=−∞2^𝑘𝑝𝜆⎞⎟⎟⎠^1/𝑝‖𝑓‖_𝑀̇𝐾^{𝛼,𝜆𝑝,𝑞}=𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^𝑞_(ℝ^𝑛₎‖𝑓‖_𝑀̇𝐾^{𝛼,𝜆𝑝,𝑞}_(ℝ𝑛).(3.23)
Similarly to the proof of 𝐸₁, we have 𝐸₂𝐸≤₃𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO^{𝑞𝑟/(𝑞−𝑟)}_(ℝ^𝑛₎‖𝑓‖_𝑀̇𝐾^{𝛼,𝜆𝑝,𝑞}_(ℝ𝑛),≤𝐶𝐴_Φ,𝑟‖𝑏‖_Ċ‌MO¹_(ℝ^𝑛₎‖𝑓‖_𝑀̇𝐾^{𝛼,𝜆𝑝,𝑞}_(ℝ𝑛).(3.24) This finishes the proof of Theorem 2.10.

Proof of Theorem 2.11. The proof is similar to the proof of Theorem 2.10, so we omit the details.

Acknowledgment

The authors are supported in part by the NSF of China, no: 10931001, 10871173.