### Abstract

Let Formula Presented the weight function on Formula Presented denote the space of even integrable functions on ℝ with respect to Formula Presented and define the radial maximal operator Formula Presented, as usual. We introduce a real Hardy space H^{1}(Δ) as the set of all even locally integrable functions f on ℝ whose radial maximal function Formula Presented belongs to L^{1}(Δ). We shall obtain a relation between H^{1}(Δ) and H^{1}(ℝ). As an application of H^{1}(Δ), we shall consider (H^{1}(Δ),L^{1}(Δ)) boundedness of integral operators associated to the Poisson kernel for Jacobi analysisj the Poisson maximal operator Formula Presented, the Littlewood-Paley g-function, and the Lusin area function S. They are bounded on L^{p}(Δ) for p >1, but not true for p = 1. We shall prove that Formula Presented, g, and a modified Formula Presented are bounded form H^{1}(Δ) to L^{1}(Δ).

Original language | English |
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Title of host publication | Infinite Dimensional Harmonic Analysis IV: On the Interplay Between Representation Theory, Random Matrices, Special Functions, and Probability |

Publisher | World Scientific Publishing Co. |

Pages | 158-171 |

Number of pages | 14 |

ISBN (Electronic) | 9789812832825 |

ISBN (Print) | 9812832815, 9789812832818 |

DOIs | |

Publication status | Published - 2008 Jan 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Infinite Dimensional Harmonic Analysis IV: On the Interplay Between Representation Theory, Random Matrices, Special Functions, and Probability*(pp. 158-171). World Scientific Publishing Co.. https://doi.org/10.1142/9789812832825_0010