### Abstract

For α ≥ β ≥ -1/2 let denote the weight function on R_{+} and L^{1}(δ) the space of integrable functions on R_{+} with respect to δ(x)dx, equipped with a convolution structure. For a suitable φ ∈ L^{1}(δ), we put for t > 0 and define the radial maximal operator M_{φ} as usual manner. We introduce a real Hardy space H^{1}(δ) as the set of all locally integrable functions f on R_{+} whose radial maximal function M_{φ}(f) belongs to L^{1}(δ). In this paper we obtain a relation between H^{1}(δ) and H^{1}(R). Indeed, we characterize H^{1}(δ) in terms of weighted H^{1} Hardy spaces on R via the Abel transform of f. As applications of H^{1}(δ) and its characterization, we shall consider (H^{1}(δ),L^{1}(δ))-boundedness of some operators associated to the Poisson kernel for Jacobi analysis: the Poisson maximal operator M_{P}, 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. Instead, M_{P, g} and a modified S_{a,γ} are bounded from H^{1}(δ) to L^{1}(δ).

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

Pages (from-to) | 201-229 |

Number of pages | 29 |

Journal | Analysis in Theory and Applications |

Volume | 25 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 Sep |

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### Keywords

- Hardy space
- Jacobi analysis
- Littlewood-Paley function
- Lusin function

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**H ^{1}-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis.** / Kawazoe, Takeshi.

Research output: Contribution to journal › Article

^{1}-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis',

*Analysis in Theory and Applications*, vol. 25, no. 3, pp. 201-229. https://doi.org/10.1007/s10496-009-0201-1

}

TY - JOUR

T1 - H1-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis

AU - Kawazoe, Takeshi

PY - 2009/9

Y1 - 2009/9

N2 - For α ≥ β ≥ -1/2 let denote the weight function on R+ and L1(δ) the space of integrable functions on R+ with respect to δ(x)dx, equipped with a convolution structure. For a suitable φ ∈ L1(δ), we put for t > 0 and define the radial maximal operator Mφ as usual manner. We introduce a real Hardy space H1(δ) as the set of all locally integrable functions f on R+ whose radial maximal function Mφ(f) belongs to L1(δ). In this paper we obtain a relation between H1(δ) and H1(R). Indeed, we characterize H1(δ) in terms of weighted H1 Hardy spaces on R via the Abel transform of f. As applications of H1(δ) and its characterization, we shall consider (H1(δ),L1(δ))-boundedness of some operators associated to the Poisson kernel for Jacobi analysis: the Poisson maximal operator MP, 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. Instead, MP, g and a modified Sa,γ are bounded from H1(δ) to L1(δ).

AB - For α ≥ β ≥ -1/2 let denote the weight function on R+ and L1(δ) the space of integrable functions on R+ with respect to δ(x)dx, equipped with a convolution structure. For a suitable φ ∈ L1(δ), we put for t > 0 and define the radial maximal operator Mφ as usual manner. We introduce a real Hardy space H1(δ) as the set of all locally integrable functions f on R+ whose radial maximal function Mφ(f) belongs to L1(δ). In this paper we obtain a relation between H1(δ) and H1(R). Indeed, we characterize H1(δ) in terms of weighted H1 Hardy spaces on R via the Abel transform of f. As applications of H1(δ) and its characterization, we shall consider (H1(δ),L1(δ))-boundedness of some operators associated to the Poisson kernel for Jacobi analysis: the Poisson maximal operator MP, 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. Instead, MP, g and a modified Sa,γ are bounded from H1(δ) to L1(δ).

KW - Hardy space

KW - Jacobi analysis

KW - Littlewood-Paley function

KW - Lusin function

UR - http://www.scopus.com/inward/record.url?scp=70350230080&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350230080&partnerID=8YFLogxK

U2 - 10.1007/s10496-009-0201-1

DO - 10.1007/s10496-009-0201-1

M3 - Article

AN - SCOPUS:70350230080

VL - 25

SP - 201

EP - 229

JO - Analysis in Theory and Applications

JF - Analysis in Theory and Applications

SN - 1672-4070

IS - 3

ER -