Spectral Analysis and Geometry of Sub-Laplacian and Related Grushin-type Operators

Bauer, Wolfram; Furutani, Kenro; Iwasaki, Chisato

doi:10.1007/978-3-0348-0024-2_4

Cited by 8 publications

(22 citation statements)

References 44 publications

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“…As can be seen from the general formula in Theorem 4.2 and more concretely from earlier results in [3,8] and Section 5 the spectral zeta functions of these sub-Laplacians have a common term. In this appendix we study its analytic behavior in the simplest case.…”

Section: Appendix : a General Formulamentioning

confidence: 88%

“…In this section we treat typical examples which were previously studied in [3,7,8]. In order to cover all low dimensional cases (dim ≤ 6) we list them here again.…”

Section: Heat Trace Expression: Concrete Casesmentioning

confidence: 98%

“…Further details on the discussion below can be found in Section 6 of [7] or Sections 13 and 14 of [8]. We consider a sub-Riemannian structure H on F (3+3) generated by the three left invariant vector…”

Section: Six-dimensional Free Nilmanifoldmentioning

confidence: 99%

“…In our earlier papers [3,8] and [7] we determined the spectral zeta function of the (sub)-Laplacian for low dimensional nilmanifolds. Here we deal with a class of nilmanifolds L\G associated to pseudo H-type algebras which will be defined in Section 6.…”

Section: Introductionmentioning

confidence: 99%

“…In order to have more concrete examples we provide heat traces and spectral zeta functions in an explicit form for low dimensional (dim ≤ 6) nilmanifolds in Section 5 following [3,7,8].…”

Section: Introductionmentioning

confidence: 99%

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Spectral zeta function on pseudo H-type nilmanifolds

Bauer

Furutani

Iwasaki

2015

Indian J Pure Appl Math

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We explain the explicit integral form of the heat kernel for the sub-Laplacian on two step nilpotent Lie groups G based on the work of Beals, Gaveau and Greiner. Using such an integral form we study the heat trace of the sub-Laplacian on nilmanifolds L\G where L is a lattice. As an application a common property of the spectral zeta function for the sub-Laplacian on L\G is observed. In particular, we introduce a special class of nilpotent Lie groups, called pseudo H-type groups which are generalizations of groups previously considered by Kaplan. As is known such groups always admit lattices. Here we aim to explicitly calculate the heat trace and the spectrum of the (sub)-Laplacian on various low dimensional compact nilmanifolds including several pseudo H-type nilmanifolds L\G, i.e. where G is a pseudo H-type group. In an appendix we discuss a zeta function which typically appears as the Mellin transform for these heat traces.

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Section: Appendix : a General Formulamentioning

confidence: 88%

“…In this section we treat typical examples which were previously studied in [3,7,8]. In order to cover all low dimensional cases (dim ≤ 6) we list them here again.…”

Section: Heat Trace Expression: Concrete Casesmentioning

confidence: 98%

Section: Six-dimensional Free Nilmanifoldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In order to have more concrete examples we provide heat traces and spectral zeta functions in an explicit form for low dimensional (dim ≤ 6) nilmanifolds in Section 5 following [3,7,8].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Spectral zeta function on pseudo H-type nilmanifolds

Bauer

Furutani

Iwasaki

2015

Indian J Pure Appl Math

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Complete classification of pseudo H-type Lie algebras: I

Furutani

Markina

2017

Geom Dedicata

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Let N be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product . , . and let N = V ⊕ ⊥ Z, where Z is the centre of the Lie algebra and V its orthogonal complement with respect to the scalar product. We study the classification of the Lie algebras for which the space V arises as a representation space of a Clifford algebra Cl(R r,s ) and the representation map J : Cl(R r,s ) → End(V ) is related to the Lie algebra structure by Jzv, w = z, [v, w] for all z ∈ R r,s and v, w ∈ V . The classification is based on the range of parameters r and s and is completed for the Clifford modules V , having minimal possible dimension, that are not necessary irreducible. We find the necessary condition for the existence of a Lie algebra isomorphism according to the range of integer parameters 0 ≤ r, s < ∞. We present the constructive proof for the isomorphism map for isomorphic Lie algebras and defined the class of non-isomorphic Lie algebras.1 2 KENRO FURUTANI, IRINA MARKINA Lie algebras. For the Clifford algebras Cl(R r,0 ), generated by the Euclidean space R r , the H-type algebras N r,0 (V ) were introduced by Kaplan [24] and attracted a lot of attention [6,10,13,25,26,34,35]. The Lie algebras N r,0 (V ) is a typical example of a standard metric form. For the Clifford algebras Cl(R r,s ), generated by a sign indefinite non-degenerate scalar product space R r,s , the pseudo H-type Lie algebras N r,s (V ) were introduced in [11], and studied in [12,16,18,20,21].We study the isomorphism properties between the Lie algebras N r,s (V ). We show that the Lie algebras N r,s (V ) can not be isomorphic to N u,t (V ) unless r = t and s = u or r = u and s = t. The present paper is the first part of the complete classification, where we concentrate on the classification of the Lie algebras based on the Clifford modules of minimal possible dimensions (which are not necessarily irreducible), admitting a scalar product making the representation map J z skew symmetric. Then, by making use the Atiyah-Bott periodicity for underlying Clifford algebras we extend the study to an arbitrary dimension pseudo H-type algebras. We also show that the Lie algebras based on the non-equivalent irreducible Clifford modules are isomorphic. We stress, that the isomorphic relation between the Clifford algebras and the associated pseudo H-type Lie algebras is not functorial. In some cases the isomorphic Clifford algebras lead to isomorphic Lie algebras, in other cases not.Apart from being motivated by itself interesting mathematical question of the classification of Lie algebras, we want to mention here possible applications in other areas of mathematics. It was shown [10,16] that the pseudo H-type Lie algebras admit the integer structure constants that in its turn, according to the Malćev theorem [31], guarantees the existence of lattices on the corresponding Lie groups. The factorization of pseudo H-type Lie groups by lattices gives a vast of new examples of nilmanifolds, which type strongly depend on the classification of pseudo H-type alg...

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From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

Casarino

Ciatti

Martini

2019

Advances in Mathematics

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We prove a sharp multiplier theorem of Mihlin-Hörmander type for the Grushin operator on the unit sphere in R 3 , and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics.2010 Mathematics Subject Classification. 33C55, 42B15 (primary); 53C17, 58J50 (secondary).So the system of vector fields {Z 1 , Z 2 } is 2-step bracket-generating and determines a sub-Riemannian structure on S (more on sub-Riemannian geometry can be found, e.g., in [BeRi, CaCh, Mo]). At each point z ∈ S, the horizontal distributionis given the inner product ·, · z corresponding to the normfor all z ∈ S and v ∈ H z S. Note that the horizontal distribution has not constant rank and degenerates at the equator E = {z ∈ S : z 3 = 0}. If z ∈ E, then dim H z S = 1 and | · | z coincides with (the restriction of) the standard Riemannian norm. If z ∈ S \ E, then dim H z S = 2 and {Z 1 | z , Z 2 | z } is an orthonormal basis of H z S with respect to the inner product ·, · z .

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Spectral Analysis and Geometry of Sub-Laplacian and Related Grushin-type Operators

Cited by 8 publications

References 44 publications

Spectral zeta function on pseudo H-type nilmanifolds

Spectral zeta function on pseudo H-type nilmanifolds

Complete classification of pseudo H-type Lie algebras: I

From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

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