### Abstract

Let (ℝ _{+}; Δ(x)dx) be a Jacobi hypergroup with weight function Δ(x) = c(sinhx) ^{2α+1}.coshx) ^{2β+1}. As in the Euclidean case, the real Hardy space H ^{1}(Δ) for (ℝ _{+}; Δ(x)dx) is defined as the set of all locally integrable functions on ℝ _{+} whose radial maximal functions belong to L ^{1}(Δ). In this paper we give a characterization of H ^{1}(Δ) in terms of weighted Triebel-Lizorkin spaces on ℝ via the Abel transform. As an application, we introduce three types of atoms for (ℝ _{+}; Δ), one of them is smooth, and give an atomic decomposition of H ^{1}(Δ).

Original language | English |
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Pages (from-to) | 389-404 |

Number of pages | 16 |

Journal | Advances in Pure and Applied Mathematics |

Volume | 2 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2011 Sep |

### Fingerprint

### Keywords

- Atomic decomposition
- Hardy space
- Jacobi analysis

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Atomic decomposition of a real Hardy space for Jacobi analysis.** / Kawazoe, Takeshi.

Research output: Contribution to journal › Article

*Advances in Pure and Applied Mathematics*, vol. 2, no. 3-4, pp. 389-404. https://doi.org/10.1515/APAM.2010.036

}

TY - JOUR

T1 - Atomic decomposition of a real Hardy space for Jacobi analysis

AU - Kawazoe, Takeshi

PY - 2011/9

Y1 - 2011/9

N2 - Let (ℝ +; Δ(x)dx) be a Jacobi hypergroup with weight function Δ(x) = c(sinhx) 2α+1.coshx) 2β+1. As in the Euclidean case, the real Hardy space H 1(Δ) for (ℝ +; Δ(x)dx) is defined as the set of all locally integrable functions on ℝ + whose radial maximal functions belong to L 1(Δ). In this paper we give a characterization of H 1(Δ) in terms of weighted Triebel-Lizorkin spaces on ℝ via the Abel transform. As an application, we introduce three types of atoms for (ℝ +; Δ), one of them is smooth, and give an atomic decomposition of H 1(Δ).

AB - Let (ℝ +; Δ(x)dx) be a Jacobi hypergroup with weight function Δ(x) = c(sinhx) 2α+1.coshx) 2β+1. As in the Euclidean case, the real Hardy space H 1(Δ) for (ℝ +; Δ(x)dx) is defined as the set of all locally integrable functions on ℝ + whose radial maximal functions belong to L 1(Δ). In this paper we give a characterization of H 1(Δ) in terms of weighted Triebel-Lizorkin spaces on ℝ via the Abel transform. As an application, we introduce three types of atoms for (ℝ +; Δ), one of them is smooth, and give an atomic decomposition of H 1(Δ).

KW - Atomic decomposition

KW - Hardy space

KW - Jacobi analysis

UR - http://www.scopus.com/inward/record.url?scp=84858394238&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858394238&partnerID=8YFLogxK

U2 - 10.1515/APAM.2010.036

DO - 10.1515/APAM.2010.036

M3 - Article

AN - SCOPUS:84858394238

VL - 2

SP - 389

EP - 404

JO - Advances in Pure and Applied Mathematics

JF - Advances in Pure and Applied Mathematics

SN - 1867-1152

IS - 3-4

ER -