## Necessary and sufficient conditions for the boundedness of Dunkl-type fractional maximal operator in the Dunkl-type Morrey spaces.(English)Zbl 1210.42035

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.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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### References:

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