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

• Main Authors: Emin Guliyev, Ahmet Eroglu, Yagub Mammadov
• Format: Article
• Language: English
• Published: Hindawi Limited 2010-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://dx.doi.org/10.1155/2010/976493

We consider the generalized shift operator, associated with the Dunkl operator Λ𝛼(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 𝑀𝛽 in the Dunkl-type Morrey space 𝐿𝑝,𝜆,𝛼(ℝ), 0≤𝜆<2𝛼+2. We obtain necessary and sufficient conditions on the parameters for the boundedness 𝑀𝛽, 0≤𝛽<2𝛼+2 from the spaces 𝐿𝑝,𝜆,𝛼(ℝ) to the spaces 𝐿𝑞,𝜆,𝛼(ℝ), 1<𝑝≤𝑞<∞, and from the spaces 𝐿1,𝜆,𝛼(ℝ) to the weak spaces 𝑊𝐿𝑞,𝜆,𝛼(ℝ), 1<𝑞<∞. As an application of this result, we get the boundedness of 𝑀𝛽 from the Dunkl-type Besov-Morrey spaces 𝐵𝑠𝑝𝜃,𝜆,𝛼(ℝ) to the spaces 𝐵𝑠𝑞𝜃,𝜆,𝛼(ℝ), 1<𝑝≤𝑞<∞, 0≤𝜆<2𝛼+2, 1/𝑝−1/𝑞=𝛽/(2𝛼+2−𝜆), 1≤𝜃≤∞, and 0<𝑠<1.