## Abstract

Fix λ > 0. Consider the Hardy space H

(equations present)

on R

(a) a weak factorization of H

(b) that the BMO space associated with S

^{1}(R_{+}, dm_{λ}) in the sense of Coifman and Weiss, where R_{+ }:= (0,∞) and dm_{λ}:= x^{2λ}dx with dx the Lebesgue measure. Also, consider the Bessel operators(equations present)

on R

_{+}. The Hardy spaces H^{1}_{Δλ}and H^{1}_{Sλ}associated with Δ_{λ}and S_{λ}are defined via the Riesz transforms R_{Δλ}:= ∂x(Δ_{λ})^{−1/2}and R_{Sλ}:= x^{λ}∂_{x}x^{−λ}(S_{λ})^{−1/2}, respectively. It is known that H^{1}_{Δ λ}and H^{1}(R_{+}, dm_{λ}) coincide but that they are different from H^{1}_{Sλ}. In this article, we prove the following:(a) a weak factorization of H

^{1}(R_{+}, dm_{λ}) by using a bilinear form of the Riesz transform R_{Δλ}, which implies the characterization of the BMO space associated with Δ_{λ}via the commutators related to RΔ_{λ};(b) that the BMO space associated with S

_{λ}cannot be characterized by commutators elated to R_{Sλ}, which implies that H^{1}_{Sλ}does not have a weak factorization via a bilinear form of the Riesz transform R_{Sλ}.Original language | English |
---|---|

Pages (from-to) | 1081-1106 |

Number of pages | 26 |

Journal | Indiana University Mathematics Journal |

Volume | 66 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2017 |

## Keywords

- BMO
- commutator
- Hardy space
- factorization
- Bessel operator
- Riesz transform