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

It is known that the Dunkl-type fractional integral operator $I_{\beta }$ $(0<\beta <2\alpha +2=d_{\alpha })$ is bounded from $L^p(\mathbb{R},d{\mu }_{\alpha })$ to $L^q(\mathbb{R},d{\mu }_{\alpha })$ 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 $BM{O}_{\phi }$ 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.