Riesz transforms in one dimension

Andrew Hassell*, Adam Sikora

*Corresponding author for this work

    Research output: Contribution to journalArticle

    8 Citations (Scopus)

    Abstract

    We study the boundedness on Lp of the Riesz transform δL-1/2, where I is one of several operators defined on ℝ or ℝ+, endowed with the measure rd-1 dr, d > 1, where dr is Lebesgue measure. For integer d, this mimics the measure on Euclidean d-dimensional space, and in this case our setup is equivalent to looking at the Laplacian acting on radial functions on Euclidean space or variations of Euclidean space such as the exterior of a sphere (with either Dirichlet or Neumann boundary conditions), or the connected sum of two copies of ℝd. In this way we illuminate some recent results on the Riesz transform on asymptotically Euclidean manifolds. We are however interested in all real values of d > 1, and another goal of our analysis is to study the range of boundedness as a function of d; it is particularly interesting to see the behaviour as d crosses 2. For example, in one of our cases which models radial functions on the connected sum of two copies of ℝd, the upper threshold for Lp boundedness is p=d for d ≥ 2 and p=d/(d-1) for d < 2. Only in the case d=2 is the Riesz transform actually bounded on Lp when p is equal to the upper threshold. We also study the Riesz transform when we have an inverse square potential, or a delta function potential; these cases provide a simple model for recent results of the first author and Guillarmou. Finally we look at the Hodge projector in a slightly more general setup. Indiana University Mathematics Journal

    Original languageEnglish
    Pages (from-to)823-852
    Number of pages30
    JournalIndiana University Mathematics Journal
    Volume58
    Issue number2
    DOIs
    Publication statusPublished - 2009

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