## On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line.(English)Zbl 1162.42006

The Dunkl operators are differential-difference operators associated with the reflection group $$Z_2$$ on $$\mathbb{R}^1$$. F. Soltani [J. Funct. Anal. 221, No. 1, 205–225 (2005; Zbl 1071.42008)] considered the maximal operator associated with the Dunkl operators. The authors study the fractional maximal function and the fractional integral associated with the Dunkl operators.
Let $$\alpha > -1/2$$ and $$\mu_{\alpha}$$ be the weighted Lebesgue measure on $$\mathbb{R}^1$$ given by $$d\mu_{\alpha}(x) = (2^{\alpha + 1} \Gamma (\alpha + 1))^{-1} | x |^{2 \alpha +1}dx,$$ and $$L_{p, \alpha}(\mathbb{R}^1)$$ is the space of functions $$f$$ such that $$\| f \|_{p, \alpha}= \left( \int_{\mathbb{R}^1} | f(x) |^{p} d \mu_{\alpha}(x) \right)^{1/p}< \infty.$$ The operators $$\tau_x, x \in \mathbb{R}^1$$ are Dunkl translation operators, and the Dunkl-type fractional integral is defined by $I_{\beta}f(x) = \int_{\mathbb{R}^1} \tau_x | y |^{\beta - 2\alpha -2}f(y) d \mu_{\alpha}(y) \quad 0 < \beta < 2\alpha +2.$ They prove the following: If $$1<p< \frac{2\alpha +2}{\beta}$$, then the condition $$\frac{1}{p} = \frac{1}{q}- \frac{\beta}{2\alpha +2}$$ is the necessary and sufficient for the boundedness of $$I_{\beta}$$ from $$L_{p, \alpha}$$ to $$L_{q, \alpha}$$.
They also consider the estimates on BMO and the Besov spaces.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces

Zbl 1071.42008
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### References:

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