### 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 |
---|---|

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

**Real hardy space for jacobi analysis and its applications.** / Kawazoe, Takeshi.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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

}

TY - CHAP

T1 - Real hardy space for jacobi analysis and its applications

AU - Kawazoe, Takeshi

PY - 2008/1/1

Y1 - 2008/1/1

N2 - 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 H1(Δ) as the set of all even locally integrable functions f on ℝ whose radial maximal function Formula Presented belongs to L1(Δ). We shall obtain a relation between H1(Δ) and H1(ℝ). As an application of H1(Δ), we shall consider (H1(Δ),L1(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 Lp(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 H1(Δ) to L1(Δ).

AB - 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 H1(Δ) as the set of all even locally integrable functions f on ℝ whose radial maximal function Formula Presented belongs to L1(Δ). We shall obtain a relation between H1(Δ) and H1(ℝ). As an application of H1(Δ), we shall consider (H1(Δ),L1(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 Lp(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 H1(Δ) to L1(Δ).

UR - http://www.scopus.com/inward/record.url?scp=84971254611&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84971254611&partnerID=8YFLogxK

U2 - 10.1142/9789812832825_0010

DO - 10.1142/9789812832825_0010

M3 - Chapter

AN - SCOPUS:84971254611

SN - 9812832815

SN - 9789812832818

SP - 158

EP - 171

BT - Infinite Dimensional Harmonic Analysis IV: On the Interplay Between Representation Theory, Random Matrices, Special Functions, and Probability

PB - World Scientific Publishing Co.

ER -