### Abstract

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L ^{2}((0, ∞), dt/t), γ(H,B) represents the space of γ-radonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BMO _{L}(R ^{n},B) (respectively, H _{L} ^{1}(R _{n},B)) into BMO _{L}(R ^{n},γ(H,B)) (respectively, H _{L} ^{1}(R ^{n},γ(H,B))), where BMO _{L} and H _{L} ^{1} denote BMO and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO _{L}(R ^{n},B) and H _{L} ^{1}(R _{n},B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.

Original language | English |
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Pages (from-to) | 3804-3856 |

Number of pages | 53 |

Journal | Journal of Functional Analysis |

Volume | 263 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 15 2012 |

### Keywords

- BMO
- Hardy spaces
- Hermite operator
- Littlewood-Paley-Stein functions
- UMD Banach spaces
- γ-Radonifying operators

### ASJC Scopus subject areas

- Analysis

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## Cite this

*Journal of Functional Analysis*,

*263*(12), 3804-3856. https://doi.org/10.1016/j.jfa.2012.09.010