Necessary and sufficient conditions for the boundedness of Dunkl-type fractional maximal operator in the Dunkl-type Morrey spaces (Q1957604)

The authors consider Morrey type spaces (\(\equiv\) Dunkl-type Morrey spaces) \(L_{p,\lambda,\alpha},\) associated with the Dunkl operator, defined by the norm \[ \|f\|_{p,\lambda,\alpha}:= \sup\limits_{x\in\mathbb{R},r>0}\left(\frac{1}{r^\lambda} \int_{|y|<r}\tau_x|f(y)|^p d\mu_\alpha(y)\right)^\frac{1}{p}, \;\;d\mu_\alpha(y)= \text{const} |y|^{2\alpha+1}dy, \] where \(\tau_x\) is the Dunkl translation operator, and the corresponding weak Dunkl-type Morrey spaces. The main result is the theorem which provides condition for the boundedness of the maximal fractional operator \(M_\beta\) (introduced in similar terms via \(\tau_x\)) from \(L_{p,\lambda,\alpha}\) to \(L_{q,\lambda,\alpha}\), when \(1\leq p\leq\frac{2\alpha+2-\lambda}{\beta}\) with the weak version of the space \(L_{q,\lambda,\alpha}\) when \(p=1.\) In the case \(1\leq p<\frac{2\alpha+2-\lambda}{\beta}\) the obtained condition \[ \frac{1}{p}-\frac{1}{q}=\frac{\beta}{2\alpha+2-\lambda} \] is necessary and sufficient.