Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

Abstract

We consider the generalized shift operator, associated with the Dunkl operator Λ_𝛼(𝑓)(𝑥)=(𝑑/𝑑𝑥)𝑓(𝑥)+((2𝛼+1)/𝑥)((𝑓(𝑥)−𝑓(−𝑥))/2), 𝛼>−1/2. We study the boundedness of the Dunkl-type fractional maximal operator 𝑀_𝛽 in the Dunkl-type Morrey space 𝐿_{𝑝,𝜆,𝛼}(ℝ), 0≤𝜆<2𝛼+2. We obtain necessary and sufficient conditions on the parameters for the boundedness 𝑀_𝛽, 0≤𝛽<2𝛼+2 from the spaces 𝐿_{𝑝,𝜆,𝛼}(ℝ) to the spaces 𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑝≤𝑞<∞, and from the spaces 𝐿_1,𝜆,𝛼(ℝ) to the weak spaces 𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑞<∞. As an application of this result, we get the boundedness of 𝑀_𝛽 from the Dunkl-type Besov-Morrey spaces 𝐵^{𝑠𝑝𝜃,𝜆,𝛼}(ℝ) to the spaces 𝐵^{𝑠𝑞𝜃,𝜆,𝛼}(ℝ), 1<𝑝≤𝑞<∞, 0≤𝜆<2𝛼+2, 1/𝑝−1/𝑞=𝛽/(2𝛼+2−𝜆), 1≤𝜃≤∞, and 0<𝑠<1.

1. Introduction

On the real line, the Dunkl operators Λ_𝛼 are differential-difference operators introduced in 1989 by Dunkl [1]. For a real parameter 𝛼>−1/2, we consider the Dunkl operator, associated with the reflection group ℤ₂ on ℝ: Λ_𝛼(𝑓)(𝑥)∶=𝑑𝑑𝑥𝑓(𝑥)+2𝛼𝑥+1𝑓(𝑥)−2𝑓(−𝑥).(1.1)

In the theory of partial differential equations, together with weighted 𝐿_𝑝,𝑤(ℝ^𝑛) spaces, Morrey spaces 𝐿_𝑝,𝜆(ℝ^𝑛) play an important role. Morrey spaces were introduced by Morrey in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations (see [2]).

The Hardy-Littlewood maximal function, fractional maximal function, and fractional integrals are important technical tools in harmonic analysis, theory of functions, and partial differential equations. In the works [3–5], the maximal operator and in [6, 7] the fractional maximal operator associated with the Dunkl operator on ℝ were studied. In this work, we study the boundedness of the fractional maximal operator 𝑀_𝛽 (Dunkl-type fractional maximal operator) in Morrey spaces 𝐿_{𝑝,𝜆,𝛼}(ℝ) (Dunkl-type Morrey spaces) associated with the Dunkl operator on ℝ. We obtain the necessary and sufficient conditions for the boundedness of the operator 𝑀_𝛽 from the spaces 𝐿_{𝑝,𝜆,𝛼}(ℝ) to 𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑝≤𝑞<∞, and from the spaces 𝐿_1,𝜆,𝛼(ℝ) to the weak spaces 𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑞<∞.

The paper is organized as follows. In Section 2, we present some definitions and auxiliary results. In Section 3, we give our main result on the boundedness of the operator 𝑀_𝛽 in 𝐿_{𝑝,𝜆,𝛼}(ℝ). We obtain necessary and sufficient conditions on the parameters for the boundedness of the operator 𝑀_𝛽 from the spaces 𝐿_{𝑝,𝜆,𝛼}(ℝ) to the spaces 𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑝≤𝑞<∞, and from the spaces 𝐿_1,𝜆,𝛼(ℝ) to the weak spaces 𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑞<∞. As an application of this result, in Section 4 we prove the boundedness of the operator 𝑀_𝛽 from the Dunkl-type Besov-Morrey spaces 𝐵^{𝑠𝑝𝜃,𝜆,𝛼}(ℝ) to the spaces 𝐵^{𝑠𝑞𝜃,𝜆,𝛼}(ℝ), 1<𝑝≤𝑞<∞, 0≤𝜆<2𝛼+2, 1/𝑝−1/𝑞=𝛽/(2𝛼+2−𝜆), 1≤𝜃≤∞, and 0<𝑠<1.

Finally, we mention that, 𝐶 will be always used to denote a suitable positive constant that is not necessarily the same in each occurrence.

2. Preliminaries

Let 𝛼>−1/2 be a fixed number and 𝜇_𝛼 be the weighted Lebesgue measure on ℝ, given by 𝑑𝜇_𝛼(𝑥)∶=2^𝛼+1Γ(𝛼+1)⁻¹|𝑥|^2𝛼+1𝑑𝑥.(2.1)

For every 1≤𝑝≤∞, we denote by 𝐿_𝑝,𝛼(ℝ)=𝐿_𝑝(𝑑𝜇_𝛼)(ℝ) the spaces of complex-valued functions 𝑓, measurable on ℝ such that ‖𝑓‖_𝑝,𝛼∶=‖𝑓‖_ℝ∞,𝛼||𝑓(𝑥)||^𝑝𝑑𝜇_𝛼(𝑥)^1/𝑝<∞if𝑝∈[1,∞),∶=esssup_𝑥∈ℝ||𝑓(𝑥)||if𝑝=∞.(2.2)

For 1≤𝑝<∞ we denote by 𝑊𝐿_𝑝,𝛼(ℝ), the weak 𝐿_𝑝,𝛼(ℝ) spaces defined as the set of locally integrable functions 𝑓(𝑥), 𝑥∈ℝ with the finite norm ‖𝑓‖_{𝑊𝐿𝑝,𝛼}∶=sup_𝑟>0𝑟𝜇_𝛼𝑥∈ℝ∶||𝑓(𝑥)||>𝑟^1/𝑝.(2.3)

Note that 𝐿_𝑝,𝛼⊂𝑊𝐿_𝑝,𝛼,‖𝑓‖_{𝑊𝐿𝑝,𝛼}≤‖𝑓‖_𝑝,𝛼∀𝑓∈𝐿_𝑝,𝛼(ℝ).(2.4)

For all 𝑥,𝑦,𝑧∈ℝ, we put 𝑊_𝛼(𝑥,𝑦,𝑧)∶=1−𝜎_{𝑥,𝑦,𝑧}+𝜎_{𝑧,𝑥,𝑦}+𝜎_{𝑧,𝑦,𝑥}Δ_𝛼(𝑥,𝑦,𝑧),(2.5) where 𝜎_{𝑥,𝑦,𝑧}∶=⎧⎪⎨⎪⎩𝑥²+𝑦²−𝑧²02𝑥𝑦iofth𝑥e,r𝑦w∈iseℝ⧵0,(2.6) and Δ_𝛼 is the Bessel kernel given by Δ_𝛼(𝑥,𝑦,𝑧)∶=⎧⎪⎨⎪⎩𝑑_𝛼|𝑥|+||𝑦||²−𝑧²𝑧²−|𝑥|−||𝑦||²^𝛼−1/2||𝑥𝑦𝑧||^2𝛼0iofth|e𝑧r|w∈is𝐴e,_𝑥,𝑦,(2.7) where 𝑑_𝛼=(Γ(𝛼+1))²/(2^𝛼−1√𝜋Γ(𝛼+1/2)) and 𝐴_𝑥,𝑦=[||𝑥|−|𝑦||,|𝑥|+|𝑦|].

In the sequel we consider the signed measure 𝜈_𝑥,𝑦, on ℝ, given by 𝜈_𝑥,𝑦∶=⎪⎪⎧⎨⎩𝑊^𝛼𝑑𝛿_𝑥(𝑥,𝑦,𝑧)𝑑𝜇_𝛼(𝑧)if𝑥,𝑦∈ℝ⧵0,𝑑𝛿_𝑦(𝑧)if𝑦=0,(𝑧)if𝑥=0.(2.8)

For 𝑥,𝑦∈ℝ and 𝑓 being a continuous function on ℝ, the Dunkl translation operator 𝜏_𝑥 is given by 𝜏_𝑥𝑓(𝑦)∶=_ℝ𝑓(𝑧)𝑑𝜈_𝑥,𝑦(𝑧).(2.9)

Using the change of variable 𝑧=Ψ(𝑥,𝑦,𝜃)=√𝑥²+𝑦²−2𝑥𝑦cos𝜃, we have also (see [8]) 𝜏_𝑥𝑓(𝑦)=𝐶_𝛼^𝜋0𝑓(Ψ)+𝑓(−Ψ)+𝑥Ψ+𝑦(𝑓(Ψ)−𝑓(−Ψ))𝑑𝜈^𝛼(𝜃),(2.10) where 𝑑𝜈_𝛼(𝜃)=(1−cos𝜃)sin^2𝛼𝜃𝑑𝜃 and 𝐶_𝛼=Γ(𝛼+1)/2√𝜋Γ(𝛼+1/2).

Proposition 2.1 (see Soltani [9]). For all 𝑥∈ℝ the operator 𝜏_𝑥 extends to 𝐿_𝑝,𝛼(ℝ), 𝑝≥1 and we have for 𝑓∈𝐿_𝑝,𝛼(ℝ), ‖‖𝜏_𝑥𝑓‖‖_{𝐿𝑝,𝛼}≤4‖𝑓‖_{𝐿𝑝,𝛼}.(2.11)

Let 𝐵(𝑥,𝑟)={𝑦∈ℝ∶|𝑦|∈]max{0,|𝑥|−𝑟}, |𝑥|+𝑟[},𝑟>0, and 𝑏_𝛼=[2^𝛼+1(𝛼+1)Γ(𝛼+1)]⁻¹. Then 𝐵(0,𝑟)=]−𝑟,𝑟[ and 𝜇_𝛼𝐵(0,𝑟)=𝑏_𝛼𝑟^2𝛼+2.

Now we define the Dunkl-type fractional maximal function (see [3–5]) by 𝑀_𝛽𝑓(𝑥)=sup_𝑟>0𝜇_𝛼𝐵(0,𝑟)^{−1+𝛽/(2𝛼+2)}_𝐵(0,𝑟)𝜏_𝑥||𝑓||(𝑦)𝑑𝜇_𝛼(𝑦),0≤𝛽<2𝛼+2.(2.12) If 𝛽=0, then 𝑀=𝑀₀ is the Dunkl-type maximal operator.

In [3–5] was proved the following theorem (see also [10]).

Theorem 2.2. (1) If 𝑓∈𝐿_1,𝛼(ℝ), then for every 𝛽>0𝜇_𝛼{𝑥∈ℝ∶𝑀𝑓(𝑥)>𝛽}≤𝐶𝛽‖𝑓‖^𝐿1,𝛼,(2.13) where 𝐶>0 is independent of 𝑓.
(2) If 𝑓∈𝐿_𝑝,𝛼(ℝ),1<𝑝≤∞, then 𝑀𝑓∈𝐿_𝑝,𝛼(ℝ) and ‖𝑀𝑓‖_{𝐿𝑝,𝛼}≤𝐶_𝑝‖𝑓‖_{𝐿𝑝,𝛼},(2.14) where 𝐶_𝑝>0 is independent of 𝑓.

Definition 2.3. Let 1≤𝑝<∞, 0≤𝜆≤2𝛼+2. We denote by 𝐿_{𝑝,𝜆,𝛼}(ℝ) Morrey space (≡ Dunkl-type Morrey space), associated with the Dunkl operator as the set of locally integrable functions 𝑓(𝑥), 𝑥∈ℝ, with the finite norm ‖𝑓‖_{𝑝,𝜆,𝛼}=sup_{𝑥∈ℝ,𝑟>0}𝑟^−𝜆_𝐵(0,𝑟)𝜏_𝑥||𝑓(𝑦)||^𝑝𝑑𝜇_𝛼(𝑦)^1/𝑝.(2.15)

Note that 𝐿_𝑝,0,𝛼(ℝ)=𝐿_𝑝,𝛼(ℝ), and if 𝜆<0 or 𝜆>2𝛼+2, then 𝐿_{𝑝,𝜆,𝛼}(ℝ)=Θ, where Θ is the set of all functions equivalent to 0 on ℝ (see also [7]).

Definition 2.4. Let 1≤𝑝<∞and 0≤𝜆≤2𝛼+2. We denote by 𝑊𝐿_{𝑝,𝜆,𝛼}(ℝ) a weak Dunkl-type Morrey space as the set of locally integrable functions 𝑓(𝑥), 𝑥∈ℝ with finite norm ‖𝑓‖_{𝑊𝐿𝑝,𝜆,𝛼}=sup_𝑡>0𝑡sup_{𝑥∈ℝ,𝑟>0}𝑟^−𝜆_{{𝑦∈𝐵(0,𝑟)∶𝜏𝑥|𝑓(𝑦)|>𝑡}}𝑑𝜇_𝛼(𝑦)^1/𝑝.(2.16)

We note that 𝐿_{𝑝,𝜆,𝛼}(ℝ)⊂𝑊𝐿_{𝑝,𝜆,𝛼}(ℝ),‖𝑓‖_{𝑊𝐿𝑝,𝜆,𝛼}≤‖𝑓‖_{𝑝,𝜆,𝛼}.(2.17)

3. Main Results

The following theorem is our main result in which we obtain the necessary and sufficient conditions for the Dunkl-type fractional maximal operator 𝑀_𝛽 to be bounded from the spaces 𝐿_{𝑝,𝜆,𝛼}(ℝ) to 𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑝<𝑞<∞ and from the spaces 𝐿_1,𝜆,𝛼(ℝ) to the weak spaces 𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ), 1<𝑞<∞.

Theorem 3.1. Let 0≤𝛽<2𝛼+2, 0≤𝜆<2𝛼+2, and 1≤𝑝≤(2𝛼+2−𝜆)/𝛽. (1)If 𝑝=1, then the condition 1−1/𝑞=𝛽/(2𝛼+2−𝜆) is necessary and sufficient for the boundedness of 𝑀_𝛽 from 𝐿_1,𝜆,𝛼(ℝ) to 𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ).(2)If 1<𝑝<(2𝛼+2−𝜆)/𝛽, then the condition (1/𝑝)−(1/𝑞)=𝛽/(2𝛼+2−𝜆) is necessary and sufficient for the boundedness of 𝑀_𝛽 from 𝐿_{𝑝,𝜆,𝛼}(ℝ) to 𝐿_{𝑞,𝜆,𝛼}(ℝ).(3)If 𝑝=(2𝛼+2−𝜆)/𝛽, then 𝑀_𝛽 is bounded from 𝐿_{𝑝,𝜆,𝛼}(ℝ) to 𝐿_∞(ℝ).

For 1≤𝑝,𝜃≤∞, 0≤𝜆<2𝛼+2, and 0<𝑠<2, the Dunkl-type Besov-Morrey 𝐵^{𝑠𝑝𝜃,𝜆,𝛼}(ℝ) consists of all functions 𝑓 in 𝐿_{𝑝,𝜆,𝛼}(ℝ) so that ‖𝑓‖_𝐵^{𝑠𝑝𝜃,𝜆,𝛼}=‖𝑓‖_{𝐿𝑝,𝜆,𝛼}+⎛⎜⎜⎝_ℝ‖‖𝜏_𝑥𝑓(⋅)−𝑓(⋅)‖‖^𝜃𝐿_{𝑝,𝜆,𝛼}|𝑥|^{2𝛼+2+𝑠𝜃}𝑑𝜇_𝛼(𝑥)⎞⎟⎟⎠^1/𝜃<∞.(3.1)

Besov spaces in the setting of the Dunkl operators were studied by Abdelkefi and Sifi [11], Bouguila et al. [12], Guliyev and Mammadov [10], and Kamoun [13]. In the following theorem, we prove the boundedness of the Dunkl-type fractional maximal operator in the Dunkl-type Besov-Morrey spaces.

Theorem 3.2. For 1<𝑝≤𝑞<∞,0≤𝜆<2𝛼+2, (1/𝑝)−(1/𝑞)=𝛽/(2𝛼+2−𝜆), 1≤𝜃≤∞, and 0<𝑠<1, the Dunkl-type fractional maximal operator 𝑀_𝛽 is bounded from 𝐵^{𝑠𝑝𝜃,𝜆,𝛼}(ℝ) to 𝐵^{𝑠𝑞𝜃,𝜆,𝛼}(ℝ). More precisely, there is a constant 𝐶>0 such that ‖‖𝑀_𝛽𝑓‖‖_{𝐵𝑠𝑞𝜃,𝜆,𝛼}≤𝐶‖𝑓‖_𝐵^{𝑠𝑝𝜃,𝜆,𝛼}(3.2) hold for all 𝑓∈𝐵^{𝑠𝑝𝜃,𝜆,𝛼}(ℝ).

Remark 3.3. Note that Theorem 3.2 in the case 𝜆=0 was proved in [10].

4. Boundedness of the Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

In the following theorem, we obtain the boundedness of the Dunkl-type fractional maximal operator 𝑀_𝛽 in the Dunkl-type Morrey spaces 𝐿_{𝑝,𝜆,𝛼}(ℝ).

Theorem 4.1. Let 0≤𝛽<2𝛼+2, 0≤𝜆<2𝛼+2, 𝑓∈𝐿_{𝑝,𝜆,𝛼}(ℝ), and 1≤𝑝≤(2𝛼+2−𝜆)/𝛽. (1)If 𝑝=1 and 1−1/𝑞=𝛽/(2𝛼+2−𝜆), then 𝑀_𝛽𝑓∈𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ) and ‖‖𝑀_𝛽𝑓‖‖_{𝑊𝐿𝑞,𝜆,𝛼}≤𝐶‖𝑓‖_1,𝜆,𝛼,(4.1) where 𝐶>0 is independent of 𝑓.(2)If 1<𝑝<(2𝛼+2−𝜆)/𝛽 and (1/𝑝)−(1/𝑞)=𝛽/(2𝛼+2−𝜆), then 𝑀_𝛽𝑓∈𝐿_{𝑞,𝜆,𝛼}(ℝ) and ‖‖𝑀_𝛽𝑓‖‖_{𝑞,𝜆,𝛼}≤𝐶‖𝑓‖_{𝑝,𝜆,𝛼},(4.2) where 𝐶>0 is independent of 𝑓.(3)If 𝑝=(2𝛼+2−𝜆)/𝛽 and 𝑞=∞, then 𝑀_𝛽𝑓∈𝐿_∞(ℝ) and ‖‖𝑀_𝛽𝑓‖‖_∞≤𝑏^{𝛼−1/𝑝(2𝛼+2)}‖𝑓‖_{𝑝,𝜆,𝛼}.(4.3)

Proof. The maximal function 𝑀𝑓(𝑥) may be interpreted as a maximal function defined on a space of homogeneous type. By this we mean a topological space 𝑋 equipped with a continuous pseudometric 𝜌 and a positive measure 𝜇 satisfying 𝜇𝐸(𝑥,2𝑟)≤𝐶₀𝜇𝐸(𝑥,𝑟)(4.4) with a constant 𝐶₀ being independent of 𝑥 and 𝑟>0. Here 𝐸(𝑥,𝑟)={𝑦∈𝑋∶𝜌(𝑥,𝑦)<𝑟},𝜌(𝑥,𝑦)=|𝑥−𝑦|. Let (𝑋,𝜌,𝜇) be a space of homogeneous type, where 𝑋=ℝ, 𝜌(𝑥,𝑦)=|𝑥−𝑦|, and 𝑑𝜇(𝑥)=𝑑𝜇_𝛼(𝑥). It is clear that this measure satisfies the doubling condition (4.4). Define 𝑀_𝜇𝑓(𝑥)=sup_𝑟>0(𝜇𝐸(𝑥,𝑟))⁻¹_{𝐸(𝑥,𝑟)}||𝑓(𝑦)||𝑑𝜇(𝑦).(4.5)
It is well known that the maximal operator 𝑀_𝜇 is bounded from 𝐿_1,𝜆(𝑋,𝜇) to 𝑊𝐿_1,𝜆(𝑋,𝜇) and is bounded on 𝐿_𝑝,𝜆(𝑋,𝜇) for 1<𝑝<∞, 0≤𝜆<2𝛼+2 (see [14, 15]).
The following inequality was proved in [6]𝑀𝑓(𝑥)≤𝐶𝑀_𝜇𝑓(𝑥),(4.6) where 𝐶>0 is independent of 𝑓.
Then from (4.6) we get the boundedness of the operator 𝑀 from 𝐿_1,𝜆,𝛼(ℝ) to 𝑊𝐿_1,𝜆,𝛼(ℝ) and on 𝐿_{𝑝,𝜆,𝛼}(ℝ), 1<𝑝<∞. Thus in the case 𝛽=0 we complete the proof of (1) and (2).
Let 𝑡>0, 0<𝛽<2𝛼+2, 𝑓∈𝐿_{𝑝,𝜆,𝛼}(ℝ), 1≤𝑝≤(2𝛼+2−𝜆)/𝛽 and (1/𝑝)−(1/𝑞)=𝛽/(2𝛼+2−𝜆). Applying the Hölders inequality we have𝑀_𝛽𝑓(𝑥)=maxsup_𝑟≥𝑡𝜇_𝛼𝐵(0,𝑟)^{𝛽/(2𝛼+2)−1}_𝐵(0,𝑟)sup_𝑟<𝑡𝜏_𝑥||𝑓(𝑦)||𝑑𝜇_𝛼(𝑦),≤𝑏_{𝛼𝛽/(2𝛼+2)}𝜇_𝛼𝐵(0,𝑟)^{𝛽/(2𝛼+2)−1}_𝐵(0,𝑟)𝜏_𝑥||𝑓(𝑦)||𝑑𝜇_𝛼(𝑦)max𝑏^{𝛼−1/𝑝}𝑡^{𝛽−(2𝛼+2−𝜆)/𝑝}‖𝑓‖_{𝑝,𝜆,𝛼},𝑡^𝛽𝑀𝑓(𝑥).(4.7)
Therefore, for all 𝑡>0, we get𝑀_𝛽𝑓(𝑥)≤𝑏^{𝛼𝛽/(2𝛼+2)}𝑏^{𝛼−1/𝑝}𝑡^{𝛽−(2𝛼+2−𝜆)/𝑝}+‖𝑓‖_{𝑝,𝜆,𝛼},𝑡^𝛽𝑀𝑓(𝑥).(4.8) The minimum value of the right-hand side (4.8) is attained at 𝑡=2𝛼+2−𝜆𝑝𝑏^{𝛼−1/𝑝}‖𝑓‖_{𝑝,𝜆,𝛼}𝑀𝑓(𝑥)^{𝑝/(2𝛼+2−𝜆)}(4.9) and hence 𝑀_𝛽𝑓(𝑥)≤𝑏^{𝛼𝛽/(2𝛼+2)−𝛽/(2𝛼+2−𝜆)}‖𝑓‖^{1−𝑝/𝑞𝑝,𝜆,𝛼}(𝑀𝑓(𝑥))^𝑝/𝑞.(4.10)
Then for 1<𝑝≤(2𝛼+2−𝜆)/𝛽 from (4.10), we have‖‖𝑀_𝛽𝑓‖‖_{𝑞,𝜆,𝛼}=sup_𝑟>0𝑟^−𝜆_𝐵(0,𝑟)𝜏_𝑥𝑀_𝛽𝑓(𝑦)^𝑞𝑑𝜇_𝛼(𝑦)^1/𝑞≤𝑏^{𝛼𝛽/(2𝛼+2)−𝛽/(2𝛼+2−𝜆)}‖𝑓‖^{1−𝑝/𝑞𝑝,𝜆,𝛼}𝑟^−𝜆_𝐵(0,𝑟)𝜏_𝑥(𝑀𝑓(𝑦))^𝑝𝑑𝜇_𝛼(𝑦)^1/𝑞≤𝑏^{𝛼𝛽/(2𝛼+2)−𝛽/(2𝛼+2−𝜆)}‖𝑓‖^{1−𝑝/𝑞𝑝,𝜆,𝛼}‖𝑀𝑓‖^{𝑝/𝑞𝑝,𝜆,𝛼}≤𝐶‖𝑓‖_{𝑝,𝜆,𝛼},(4.11) where 𝐶>0 is independent of 𝑓.
Also for 𝑝=1 from (4.10) we have‖‖𝑀_𝛽𝑓‖‖_{𝑊𝐿𝑞,𝜆,𝛼}=sup_𝑡>0𝑡sup_{𝑥∈ℝ,𝑟>0}𝑟^−𝜆_{{𝑦∈𝐵(0,𝑟)∶𝜏𝑥𝑀𝛽𝑓(𝑦)>𝑡}}𝑑𝜇_𝛼(𝑦)^1/𝑞≤sup_𝑡>0𝑡sup_{𝑥∈ℝ,𝑟>0}𝑟^−𝜆_{𝑦∈𝐵(0,𝑟)∶𝜏𝑥𝑀𝑓(𝑦)>𝑏}^{𝛼−𝛽𝑞/(2𝛼+2−𝜆)+𝛽𝑞/(2𝛼+2)}_‖𝑓‖^{1−𝑞1,𝜆,𝛼}_𝑡𝑞𝑑𝜇_𝛼(𝑦)^1/𝑞≤𝑏^{𝛼𝛽/(2𝛼+2−𝜆)−𝛽/(2𝛼+2)}‖𝑓‖^{1−1/𝑞1,𝜆,𝛼}‖𝑀𝑓‖^{1/𝑞𝑊𝐿}_1,𝜆,𝛼≤𝐶‖𝑓‖_1,𝜆,𝛼,(4.12) where 𝐶>0 is independent of 𝑓.
Therefore, the case 𝛽>0 complete the proof of (1) and (2).
(3) Let 𝑝=(2𝛼+2−𝜆)/𝛽, 𝑓∈𝐿_{𝑝,𝜆,𝛼}(ℝ); then applying Hölders inequality, we obtain𝜇_𝛼𝐵≤(0,𝜇𝑟)^{−1+𝛽/(2𝛼+2)}_𝐵(0,𝑟)𝜏_𝑥||𝑓||(𝑦)𝑑𝜇_𝛼(𝑦)_𝛼𝐵(0,𝑟)^{−1+𝛽/(2𝛼+2)+1/𝑝}_𝐵(0,𝑟)𝜏_𝑥||𝑓(𝑦)||^𝑝𝑑𝜇_𝛼(𝑦)^1/𝑝=𝑏^{𝛼−𝜆/𝑝(2𝛼+2)}𝑟^−𝜆_𝐵(0,𝑟)𝜏_𝑥||𝑓(𝑦)||^𝑝𝑑𝜇_𝛼(𝑦)^1/𝑝≤𝑏^{𝛼−𝜆/𝑝(2𝛼+2)}‖𝑓‖_{𝑝,𝜆,𝛼}.(4.13)
Thus the case 𝛽>0 completes the proof of (3).
Theorem 4.1 has been proved.

Proof of Theorem 3.1. Sufficiency part of the proof follows from Theorem 4.1.Necessity. (1) Let 1<𝑝≤(2𝛼+2−𝜆)/𝛼 and 𝑀_𝛽 be bounded from 𝐿_{𝑝,𝜆,𝛼}(ℝ) to 𝐿_{𝑞,𝜆,𝛼}(ℝ).
Define 𝑓_𝑡(𝑥)∶=𝑓(𝑡𝑥), 𝑡>0. Then‖‖𝑓_𝑡‖‖_{𝑝,𝜆,𝛼}=𝑡^{−(2𝛼+2)/𝑝}sup_{𝑥∈ℝ,𝑟>0}𝑟^−𝜆_{𝐵(0,𝑡𝑟)}𝜏_𝑡𝑥||𝑓(𝑦)||^𝑝𝑑𝜇_𝛼(𝑦)^1/𝑝=𝑡^{−(2𝛼+2−𝜆)/𝑝}‖𝑓‖_{𝑝,𝜆,𝛼}(4.14) and 𝑀_𝛽𝑓_𝑡(𝑥)=𝑡^−𝛽𝑀_𝛽𝑓(𝑡𝑥), ‖‖𝑀_𝛽𝑓_𝑡‖‖_{𝐿𝑞,𝜆,𝛼}=𝑡^−𝛽sup_{𝑥∈ℝ,𝑟>0}𝑟^−𝜆_𝐵(0,𝑟)𝜏_𝑡𝑥||𝑀_𝛽𝑓(𝑦)||^𝑞𝑑𝜇_𝛼(𝑦)^1/𝑞=𝑡^{−𝛽−(2𝛼+2)/𝑞}sup_{𝑥∈ℝ,𝑟>0}𝑟^−𝜆_{𝐵(0,𝑡𝑟)}𝜏_𝑥||𝑀_𝛽𝑓(𝑦)||^𝑞𝑑𝜇_𝛼(𝑦)^1/𝑞=𝑡^{−𝛽−(2𝛼+2−𝜆)/𝑞}‖‖𝑀_𝛽𝑓‖‖_{𝐿𝑞,𝜆,𝛼}.(4.15)
By the boundedness of 𝑀_𝛽 from 𝐿_{𝑝,𝜆,𝛼}(ℝ) to 𝐿_{𝑞,𝜆,𝛼}(ℝ),‖‖𝑀_𝛽𝑓‖‖_{𝐿𝑞,𝜆,𝛼}=𝑟^{𝛽+(2𝛼+2−𝜆)/𝑞}‖‖𝑀_𝛽𝑓_𝑟‖‖_{𝐿𝑞,𝜆,𝛼}≤𝐶𝑟^{𝛽+(2𝛼+2−𝜆)/𝑞}‖‖𝑓_𝑟‖‖_{𝑝,𝜆,𝛼}=𝐶𝑟^{𝛽+(2𝛼+2−𝜆)/𝑞−(2𝛼+2−𝜆)/𝑝}‖𝑓‖_{𝑝,𝜆,𝛼},(4.16) where 𝐶 depends only on 𝑝, 𝛽, 𝜆, and 𝛼.
If 1/𝑝>1/𝑞+𝛽/(2𝛼+2−𝜆), then for all 𝑓∈𝐿_{𝑝,𝜆,𝛼}(ℝ) we have ‖𝑀_𝛽𝑓‖_{𝑞,𝜆,𝛼}=0 as 𝑟→0, which is impossible. Similarly, if 1/𝑝<1/𝑞+𝛽/(2𝛼+2−𝜆), then for all 𝑓∈𝐿_{𝑝,𝜆,𝛼}(ℝ) we obtain ‖𝑀_𝛽𝑓‖_{𝑞,𝜆,𝛼}=0 as 𝑟→∞, which is also impossible.
Therefore, we get 1/𝑝=1/𝑞+𝛽/(2𝛼+2−𝜆).Necessity. Let 𝑀_𝛽 be bounded from 𝐿_1,𝜆,𝛼(ℝ) to 𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ). We have ‖‖𝑀_𝛽𝑓_𝑟‖‖_{𝑊𝐿𝑞,𝜆,𝛼}=𝑟^{−𝛽−(2𝛼+2−𝜆)/𝑞}‖‖𝑀_𝛽𝑓‖‖_{𝑊𝐿𝑞,𝜆,𝛼}.(4.17)
By the boundedness of 𝑀_𝛽 from 𝐿_1,𝜆,𝛼(ℝ) to 𝑊𝐿_{𝑞,𝜆,𝛼}(ℝ) it follows that‖‖𝑀_𝛽𝑓‖‖_{𝑊𝐿𝑞,𝜆,𝛼}=𝑟^{𝛽+(2𝛼+2−𝜆)/𝑞}‖‖𝑀_𝛽𝑓_𝑟‖‖_{𝑊𝐿𝑞,𝜆,𝛼}≤𝐶𝑟^{𝛽+(2𝛼+2−𝜆)/𝑞}‖‖𝑓_𝑟‖‖_1,𝜆,𝛼=𝐶𝑟^{𝛽+(2𝛼+2−𝜆)/𝑞−(2𝛼+2)}‖𝑓‖_1,𝜆,𝛼,(4.18) where 𝐶 depends only on 𝛽, 𝜆, and 𝛼.
If 1<1/𝑞+𝛽/(2𝛼+2−𝜆), then for all 𝑓∈𝐿_1,𝜆,𝛼(ℝ) we have ‖𝑀_𝛽𝑓‖_{𝑊𝐿𝑞,𝜆,𝛼}=0 as 𝑟→0. Similarly, if 1>1/𝑞+𝛽/(2𝛼+2−𝜆), then for all 𝑓∈𝐿_1,𝜆,𝛼(ℝ) we obtain ‖𝑀_𝛽𝑓‖_{𝑊𝐿𝑞,𝜆,𝛼}=0 as 𝑟→∞.
Hence we get 1=1/𝑞+𝛽/(2𝛼+2−𝜆). Thus the proof of Theorem 3.1 is completed.

Proof of Theorem 3.2. For 𝑥∈ℝ, let 𝜏_𝑥 be the generalized translation by 𝑥. By definition of the Besov spaces, it suffices to show that ‖‖𝜏_𝑥𝑀_𝛽𝑓−𝑀_𝛽𝑓‖‖_{𝐿𝑞,𝜆,𝛼}≤𝐶₂‖‖𝜏_𝑥𝑓−𝑓‖‖_{𝐿𝑝,𝜆,𝛼}.(4.19) It is easy to see that 𝜏_𝑥 commutes with 𝑀_𝛽, that is, 𝜏_𝑥𝑀_𝛽𝑓=𝑀_𝛽(𝜏_𝑥𝑓). Hence we have ||𝜏_𝑥𝑀_𝛽𝑓−𝑀_𝛽𝑓||=||𝑀_𝛽𝜏_𝑥𝑓−𝑀_𝛽𝑓||≤𝑀_𝛽||𝜏_𝑥𝑓−𝑓||.(4.20) Taking 𝐿_{𝑝,𝜆,𝛼}(ℝ) norm on both ends of the above inequality, by the boundedness of 𝑀_𝛽 from 𝐿_{𝑝,𝜆,𝛼}(ℝ) to 𝐿_{𝑞,𝜆,𝛼}(ℝ), we obtain the desired result. Theorem 3.2 is proved.

Acknowledgment

The authors express their thanks to the referee for careful reading, and helpful comments and suggestions on the manuscript of this paper.