On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line
The Dunkl operators are differential-difference operators associated with the reflection group \(Z_2\) on \(\mathbb{R}^1\). \textit{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.
scientific article; zbMATH DE number 5549442
scientific article