Riesz transforms on a class of non-doubling manifolds

Abstract

We consider a class of manifolds M obtained by taking the connected sum of a finite number of N -dimensional Riemannian manifolds of the form (R n i , δ) × (Mi, g), where Mi is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions ni are not all equal. This means that the ends have different 'asymptotic dimension', and implies that the Riemannian manifold M is not a doubling space.In the first paper in this series, by the first and third authors, we considered the case where each ni is least 3. In the present paper, we assume that one of the ni is equal to 2, which is a special and particularly interesting case. Our approach is to construct the low energy resolvent and determine the asymptotics of the resolvent kernel as the energy tends to zero. We show that the resolvent kernel (∆ + k 2 ) −1 on M has an expansion in powers of 1/ log(1/k) as k → 0, which is significantly different from the case where all ni are at least 3, in which case the expansion is in powers of k. We express the Riesz transform in terms of the resolvent to show that it is bounded on L p (M) for 1 < p ≤ 2, and unbounded for all p > 2.1991 Mathematics Subject Classification. 42B20 (primary), 47F05, 58J05 (secondary).

The Resolvent and Riesz Transform on Connected Sums of Manifolds With Different Asymptotic Dimensions

“…We express the Riesz transform in terms of the resolvent to show that it is bounded on L p for 1 < p ≤ 2. This extends results from a paper of Hassell and Sikora [2] which considered connected sums of products of Euclidean spaces with closed manifolds, where each Euclidean space has a dimension of at least three.…”

“…In the case where the negative half-line is weighted like |r| dr and the positive half-line is weighted like r d + −1 dr, d + ≥ 3, we show that the Riesz transform is L p -bounded for the range 1 < p ≤ 2, which agrees with our multidimensional case. 2 D. Nix [2]…”

Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

“…Recently, A. Hassell and A. Sikora provided another interesting example, the connected sum of a finite number of Riemannian manifolds with very strong geometric and analytic conditions. For this setting, they proved the weak type (1, 1) of the Riesz transform in [28,Theorem 7.1]. Their proof is based on spectral multipliers and the resolvent.…”

“…More precisely, for any fixed p 0 ≥ 2, there exists a Riemannian manifold of this type where the Riesz transform is bounded on L p for 2 < p < p 0 but unbounded for p > p 0 ; if p 0 > 2 it is unbounded also on L p 0 and not even of weak type ( p 0 , p 0 ). See [18,31] and [14][15][16][17][25][26][27] as well as [28] for some concrete examples. We also mention [8,9,13,20,21] and references therein, with characterizations under the assumptions of volume doubling and Gaussian heat kernel estimates.…”

Vertical and horizontal Square Functions on a Class of Non-Doubling Manifolds

“…In point of fact, the sustained interest in these model spaces has led to investigations into the boundedness of the heat maximal operators [14], the functional calculus of ∆ [6] and Littlewood-Paley decompositions [4]. Recently, and of particular significance to our article, Hassell in collaboration with the second named author considered the L p (M)-boundedness of the Riesz transforms operator ∇∆ − 1 2 on such manifolds [19]. This paper in tern is a generalisation to the non-doubling setting of the result obtained by Carron, Coulhon and Hassell in [8].…”