Riesz transforms in one dimension

Abstract

We study the boundedness on L p of the Riesz transform ∇L -1/2 , where L is one of several operators defined on R or R + , endowed with the measure r d-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 R d .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 R d , the upper threshold for L p 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 L p 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. introductionUsing elementary calculations based on modified Bessel functions, we obtain a complete description of L p -continuity properties of the Riesz transform ∇L -1/2 for several families of Laplace type operators L defined on R or R + , with respect to the measure |r| d-1 dr where dr is Lebesgue measure.This measure mimics the measure on Euclidean space R d for d = 1, 2, 3, . . .and for that reason we refer to d as the 'dimension'.However we consider all real d > 1 in this paper.We aim rather for completeness and simplicity then generality of results.However our results are a good model for considering a wide range of multidimensional Riesz transforms.In fact many surprising negative results for the Riesz transform follows from results in this note.For example,• Proposition 6.1 shows that the Riesz transform for the operator ∆ + c/|x| 2 on R d , d ≥ 3, with -(d/2 -1) 2 < c < 0 cannot be bounded on L p (R d ) unless p is in the interval

Locations

• Indiana University Mathematics Journal - View
• arXiv (Cornell University) - PDF