### 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 |
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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 1 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

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

*Analysis in Theory and Applications*,

*25*(3), 201-229. https://doi.org/10.1007/s10496-009-0201-1