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Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces
(HINDAWI PUBLISHING CORPORATION, 2010) Guliyev, Emin; Eroglu, Ahmet; Mammadov, Yagub
We consider the generalized shift operator, associated with the Dunkl operator Lambda(alpha)(f)(x) = (d/dx)f(x) + ((2 alpha + 1)/x)((f(x) - f(-x))/2), alpha > -1/2. We study the boundedness of the Dunkltype fractional maximal operator M(beta) in the Dunkl-type Morrey space L(p,lambda,alpha)(R), 0 <= lambda < 2 alpha + 2. We obtain necessary and sufficient conditions on the parameters for the boundedness M(beta), 0 <= beta < 2 alpha + 2 from the spaces L(p,lambda,alpha)(R) to the spaces L(q,lambda,alpha)(R), 1 < p <= q < infinity, and from the spaces L(1,lambda,alpha)(R) to the weak spaces WL(q,lambda,alpha)(R), 1 < q < infinity. As an application of this result, we get the boundedness of M beta from the Dunkl-type Besov-Morrey spaces Bp theta,lambda,alpha s(R) to the spaces Bq theta,lambda,alpha s(R), 1 < p <= q < infinity, 0 <= lambda < 2 alpha + 2, 1/p - 1/q = beta/(2 alpha + 2 - lambda), 1 <= theta <= infinity, and 0 < s < 1.