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

and Applied Analysis 3 For all x, y, z ∈ R, we put Wα ( x, y, z ) : ( 1 − σx,y,z σz,x,y σz,y,x ) Δα ( x, y, z ) , 2.5


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 Z 2 on R: 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 Dunkltype Besov-Morrey spaces B s pθ,λ,α R to the spaces Finally, we mention that, C will be always used to denote a suitable positive constant that is not necessarily the same in each occurrence.

Preliminaries
Let α > −1/2 be a fixed number and μ α be the weighted Lebesgue measure on R, given by 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

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 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 : and Δ α is the Bessel kernel given by Δ α x, y, z : where In the sequel we consider the signed measure ν x,y , on R, given by ν x,y :

2.8
For x, y ∈ R and f being a continuous function on R, the Dunkl translation operator τ x is given by τ x f y : R f z dν x,y z . 2.9 Using the change of variable z Ψ x, y, θ x 2 y 2 − 2xy cos θ, we have also see 8 where dν α θ 1 − cos θ sin 2α θ dθ and 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 , Then B 0, r − r, r and μ α B 0, r b α r 2α 2 .Now we define the Dunkl-type fractional maximal function see 3-5 by If β 0, then M M 0 is the Dunkl-type maximal operator.In 3-5 was proved the following theorem see also 10 . where where 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

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 .

2.16
We note that 2.17

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 < ∞. 1 If p 1, then the condition 1 − 1/q β/ 2α 2 − λ is necessary and sufficient for the boundedness of For 1 ≤ p, θ ≤ ∞, 0 ≤ λ < 2α 2, and 0 < s < 2, the Dunkl-type Besov-Morrey B s pθ,λ,α R consists of all functions f in L p,λ,α R so that Remark 3.3.Note that Theorem 3.2 in the case λ 0 was proved in 10 .
where C > 0 is independent of f.
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 ≤ C 0 μE x, r 4.4 with a constant C 0 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 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 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 .
Applying the H ölders inequality we have

4.7
Therefore, for all t > 0, we get The minimum value of the right-hand side 4.8 is attained at and hence p,λ,α Mf x p/q .4.10 Then for 1 < p ≤ 2α 2 − λ /β from 4.10 , we have where C > 0 is independent of f.Also for p 1 from 4.10 we have r −λ {y∈B 0,r : τ x M β f y >t} dμ α y where C > 0 is independent of f.Therefore, the case β > 0 complete the proof of 1 and 2 .

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.