Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres

Ahrens, Julian; Cowling, Michael; Martini, Alessio; Müller, Detlef

doi:10.1007/s00209-019-02313-w

Cited by 10 publications

(8 citation statements)

References 27 publications

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“…We consider only the case s ∈ [0, 1], since the case s ∈ [1, ∞] is completely analogous (or almost trivial when G is stratified, see 4). Define g [1] := g and, by induction,…”

Section: Proofmentioning

confidence: 99%

“…, n, and observe that d j ≥ k j for every j ∈ J thanks to Proposition 2.7. Furthermore, define Y J j as the homogeneous component of degree k j of Y (1) j for every j ∈ J , and observe that ( Y J j ) j∈J is a basis of h 0 . In addition, arguing as in 3 above we see that…”

Section: Proofmentioning

confidence: 99%

“…More precisely, they are weighted subcoercive, according to the definition introduced in [37]. 1 The starting point in the analysis of [30] is a weighted generating family X 1 , . .…”

Section: Introductionmentioning

confidence: 99%

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Functional calculus on non-homogeneous operators on nilpotent groups

Calzi

Ricci

2020

Annali di Matematica

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We study the functional calculus associated with a hypoelliptic left-invariant differential operator $$\mathcal {L}$$ L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator $$\mathcal {L}_0$$ L 0 on the ‘local’ contraction $$G_0$$ G 0 of G, as well as of the corresponding Rockland operator $$\mathcal {L}_\infty $$ L ∞ on the ‘global’ contraction $$G_\infty $$ G ∞ of G. We provide asymptotic estimates of the Riesz potentials associated with $$\mathcal {L}$$ L at 0 and at $$\infty $$ ∞ , as well as of the kernels associated with functions of $$\mathcal {L}$$ L satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for $$\mathcal {L}$$ L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with $$\mathcal {L}$$ L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

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“…We consider only the case s ∈ [0, 1], since the case s ∈ [1, ∞] is completely analogous (or almost trivial when G is stratified, see 4). Define g [1] := g and, by induction,…”

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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Functional calculus on non-homogeneous operators on nilpotent groups

Calzi

Ricci

2020

Annali di Matematica

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“…In the context of subelliptic operators on compact manifolds, "weighted spectral cluster estimates" were first obtained in the seminal work of Cowling and Sikora [CoSi] for a distinguished sub-Laplacian on SU(2), leading to a sharp multiplier theorem in that case; their technique was then applied to many different frameworks [CoKSi,CCMS,M2,ACMM]. However, the general theory developed in [CoSi], based on spectral cluster estimates involving a single weight function, does not seem to be directly applicable to the spherical Grushin operator L d,k (which, differently from the sub-Laplacian of [CoSi], is not invariant under a transitive group of isometries of the underlying manifold).…”

Section: Introductionmentioning

confidence: 99%

Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators

Casarino,

Ciatti,

Martini

2020

Preprint

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We study degenerate elliptic operators of Grushin type on the ddimensional sphere, which are singular on a k-dimensional sphere for some k < d. For these operators we prove a spectral multiplier theorem of Mihlin-Hörmander type, which is optimal whenever 2k ≤ d, and a corresponding Bochner-Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials.

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“…e.g. [2,23,27,34,46] and references therein) and applied successfully to the study of quaternionic closed operators, quaternionic function spaces and operators on them, e.g. quaternionic slice Hardy space, quaternionic de Branges space and quaternionic Hankel operator etc.…”

mentioning

confidence: 99%

Fundamental properties of Cauchy--Szegő projection on quaternionic Siegel upper half space and applications

Chang¹,

Duong²,

Li³

et al. 2021

Preprint

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We investigate the Cauchy-Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy-Szegő kernel and prove that the Cauchy-Szegő kernel is non-zero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy-Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p on quaternionic Siegel upper half space for 2/3 < p ≤ 1. Moreover, we establish the characterisation of singular values of the commutator of Cauchy-Szegő projection based on the kernel estimates and on the recent new approach by Fan-Lacey-Li. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

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Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres

Cited by 10 publications

References 27 publications

Functional calculus on non-homogeneous operators on nilpotent groups

Functional calculus on non-homogeneous operators on nilpotent groups

Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators

Fundamental properties of Cauchy--Szegő projection on quaternionic Siegel upper half space and applications

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