TY - JOUR

T1 - Riesz transforms in one dimension

AU - Hassell, Andrew

AU - Sikora, Adam

PY - 2009

Y1 - 2009

N2 - 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

AB - 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

KW - Modified bessel functions

KW - Resolvent kernels

KW - Riesz transform

UR - http://www.scopus.com/inward/record.url?scp=67249085966&partnerID=8YFLogxK

U2 - 10.1512/iumj.2009.58.3514

DO - 10.1512/iumj.2009.58.3514

M3 - Article

SN - 0022-2518

VL - 58

SP - 823

EP - 852

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 2

ER -