Factorization for hardy spaces and characterization for BMO spaces via commutators in the bessel setting

Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang

    Research output: Contribution to journalArticlepeer-review

    13 Citations (Scopus)

    Abstract

    Fix λ > 0. Consider the Hardy space H1(R+, dmλ) in the sense of Coifman and Weiss, where R+ := (0,∞) and dmλ := x dx with dx the Lebesgue measure. Also, consider the Bessel operators

    (equations present)


    on R+. The Hardy spaces H1Δλ and H1 associated with Δλ and Sλ are defined via the Riesz transforms RΔλ := ∂x(Δλ)−1/2 and R := xλxx−λ(Sλ)−1/2, respectively. It is known that H1Δ λ and H1(R+, dmλ) coincide but that they are different from H1 . In this article, we prove the following:
    (a) a weak factorization of H1(R+, dmλ) by using a bilinear form of the Riesz transform RΔλ , which implies the characterization of the BMO space associated with Δλ via the commutators related to RΔλ ;
    (b) that the BMO space associated with Sλ cannot be characterized by commutators elated to R , which implies that H1 does not have a weak factorization via a bilinear form of the Riesz transform R.
    Original languageEnglish
    Pages (from-to)1081-1106
    Number of pages26
    JournalIndiana University Mathematics Journal
    Volume66
    Issue number4
    DOIs
    Publication statusPublished - 2017

    Keywords

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

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