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Convolution operators and Bochner-Riesz means on Herz-type Hardy spaces in the Dunkl setting. (English) Zbl 1213.44003
For $T\in \mathcal{S}'(\mathbb{R})$ and $f\in \mathcal{S}(\mathbb{R})$, the Dunkl convolution product $T\ast_{\alpha}f$ is defined by $$T\ast_{\alpha}f(x)=\langle T(y), \tau_x f(-y)\rangle,\quad x\in \mathbb{R},$$ where $\tau_x$ is the Dunkl translation operator. The author investigates the Dunkl convolution operators on Herz-type Hardy spaces $\mathcal{H}_{\alpha,2}^{p}$ and establishes a version of multiplier theorem for the maximal Bochner-Riesz operators on the Herz-type Hardy spaces $\mathcal{H}_{\alpha,\infty}^{p}$. Some related results are contained in the paper by {\it S. Lu} and {\it D. Yang} [Proc. Am. Math. Soc. 126, No. 11, 3337--3346 (1998; Zbl 0905.42007)].
44A35Convolution (integral transforms)
42B15Multipliers, several variables
Full Text: DOI EuDML
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