On the Boundedness of Generalized Fractional Integral Operators in Morrey Spaces and Camapanato Spaces associated with the Dunkl Operator on the Real line

Abstract

It is known that the Dunkl-type fractional integral operator $I_\beta$ $(0 < \beta < 2\alpha + 2 =d_\alpha)$ is bounded from $L^p(\R,d\mu_\alpha)$ to $L^q (\R, d\mu_\alpha)$ when $1 < p < \frac{d_\alpha}{\beta}$ and $\frac{1}{p} - \frac{1}{q} = \frac{\beta}{d_\alpha}$. In \cite{spsa} , the authors introduced the generalized Dunkl-type fractional integral operator $T_\rho^\alpha$ and it's modified version $\tilde{T}_\rho^\alpha$ and extended the above boundedness results to the generalized Dunkl-type Morrey spaces and Dunkl-type $BMO_\phi$ spaces. In this paper we investigate the boundedness of generalized Dunkl-type fractional integral operators and it's modified version mainly on the Dunkl-type Campanato space.