### Abstract

For the Jacobi hypergroup (R+,Δ,), the weak-L1 estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution to the Euclidean convolution. More generally, let T be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood- Paley g-function for the Jacobi hypergroup, which are defined by using. Then we shall give a standard shape of Tf for f ∈ L1(Δ), from which its weak-L1 estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.

Original language | English |
---|---|

Pages (from-to) | 77-112 |

Number of pages | 36 |

Journal | Tokyo Journal of Mathematics |

Volume | 41 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 Jun 1 |

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### Keywords

- Jacobi analysis
- Kunze-Stein phenomenon
- Maximal function
- Weak-l1 estimate

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Applications of an inverse abel transform for jacobi analysis : Weak-l1 estimates and the kunze-Stein phenomenon.** / Kawazoe, Takeshi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Applications of an inverse abel transform for jacobi analysis

T2 - Weak-l1 estimates and the kunze-Stein phenomenon

AU - Kawazoe, Takeshi

PY - 2018/6/1

Y1 - 2018/6/1

N2 - For the Jacobi hypergroup (R+,Δ,), the weak-L1 estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution to the Euclidean convolution. More generally, let T be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood- Paley g-function for the Jacobi hypergroup, which are defined by using. Then we shall give a standard shape of Tf for f ∈ L1(Δ), from which its weak-L1 estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.

AB - For the Jacobi hypergroup (R+,Δ,), the weak-L1 estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution to the Euclidean convolution. More generally, let T be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood- Paley g-function for the Jacobi hypergroup, which are defined by using. Then we shall give a standard shape of Tf for f ∈ L1(Δ), from which its weak-L1 estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.

KW - Jacobi analysis

KW - Kunze-Stein phenomenon

KW - Maximal function

KW - Weak-l1 estimate

UR - http://www.scopus.com/inward/record.url?scp=85051213913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051213913&partnerID=8YFLogxK

U2 - 10.3836/tjm/1502179242

DO - 10.3836/tjm/1502179242

M3 - Article

VL - 41

SP - 77

EP - 112

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 1

ER -