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

Nix, Daniel

doi:10.1017/s0004972721000496

Cited by 2 publications

(4 citation statements)

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“…There is now a vast literature on Riesz transforms on manifolds, which we do not review here. See [1], [21] and [26] for further information and literature review.…”

Section: Introductionmentioning

confidence: 99%

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Riesz transforms on a class of non-doubling manifolds

Hassell

Sikora

2019

Communications in Partial Differential Equations

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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).

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“…There is now a vast literature on Riesz transforms on manifolds, which we do not review here. See [1], [21] and [26] for further information and literature review.…”

Section: Introductionmentioning

confidence: 99%

“…The question of stability of the boundedness of the Riesz transform under compact perturbations has been considered by Coulhon and Dungey [6] and by Jiang and Lin [24]. A more comprehensive discussion of related literature is given in the second author's PhD thesis [26].…”

Section: Introductionmentioning

confidence: 99%

Riesz transforms on a class of non-doubling manifolds

Hassell

Sikora

2019

Communications in Partial Differential Equations

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“…In the landmark article by Grigoryan and Saloff-Coste [17], the authors effectively computed, using probabilistic methods, two-sided estimates for the heat kernel generated by the Laplacian ∆ on this prototypical collection of non-doubling spaces. Although not the first to study this remarkable class of manifolds in detail (see [25] for a detailed historical account), this article acted as an inflection point for interest in this class and had a pronounced effect on the non-homogeneous community. Indeed, it essentially designated this class of manifolds as a battlefront for the advancement of non-homogeneous harmonic analysis.…”

Section: Introductionmentioning

confidence: 99%

“…We do not study the range 2M ≥ n min here. Therefore, we do not study the case n min = 2 which was investigated in [18,25].…”

Section: Introductionmentioning

confidence: 99%

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

Bailey¹,

Sikora²

2021

Preprint

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We consider a class of non-doubling manifolds M that are the connected sum of a finite number of N -dimensional manifolds of the form R n i × Mi. Following on from the work of Hassell and the second author [19], a particular decomposition of the resolvent operators (∆ + k 2 ) −M , for M ∈ N * , will be used to demonstrate that the vertical square function operatoris bounded on L p (M) for 1 < p < nmin = mini ni and weak-type (1, 1). In addition, it will be proved that the reverse inequality f p Sf p holds for p ∈ (n ′ min , nmin) and that S is unbounded for p ≥ nmin provided 2M < nmin.Similarly, for M > 1, this method of proof will also be used to ascertain that the horizontal square function operatoris bounded on L p (M) for all 1 < p < ∞ and weak-type (1, 1). Hence, for p ≥ nmin, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces H p do not coincide.

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

Cited by 2 publications

References 2 publications

Riesz transforms on a class of non-doubling manifolds

Riesz transforms on a class of non-doubling manifolds

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

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