Fundamental solution of a higher step Grushin type operator

Bauer, Wolfram; Furutani, Kenro; Iwasaki, Chisato

doi:10.1016/j.aim.2014.11.017

Cited by 25 publications

(34 citation statements)

References 18 publications

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“…The sub-Laplacian Δ G sub is positive and essentially selfadjoint in L 2 (G) when we consider it on the domain C ∞ 0 (G) (see [5] for an elementary proof of these facts including the case of higher step Grushin type operators descended from sub-Laplacians to a homogeneous space). Moreover, λ = 0 is an eigenvalue with eigenspace formed by the constant functions.…”

Section: Sub-riemannian Structure On Nilpotent Lie Groupsmentioning

confidence: 97%

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: Sub-riemannian Structure On Nilpotent Lie Groupsmentioning

confidence: 97%

Spectral zeta function on pseudo H-type nilmanifolds

Bauer

Furutani

Iwasaki

2015

Indian J Pure Appl Math

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“…This gives back a formula already obtained by Greiner [45] (see also Beals, Gaveau, Greiner [9,12]; Beals, Gaveau, Greiner, Kannai [15]; Bauer, Furutani, Iwasaki [6]). …”

Section: Examplesmentioning

confidence: 99%

“…Since our method is based on a saturation argument, we would like to highlight that a similar saturation method was also used by Beals, Gaveau, Greiner, Kannai in [15], where operators lifting to sub-Laplacians on Carnot groups of step two are considered; or in Bauer, Furutani, Iwasaki [6] (where the sub-Laplacian of the first Heisenberg group is used as a lifting a Grushin operator on R 2 ). Our theorem, allowing for general homogeneous operators, comprises both of these cases.…”

Section: Furthermore It Is Symmetricmentioning

confidence: 99%

“…Parallel to what happens in the mentioned case of L and its heat lifting H or in the case of ∆ n and ∆ n+p , the idea of obtaining a fundamental solution for the Grushin operator G via a saturation argument applied to the (explicit!) fundamental solution of G has already been exploited in the literature (e.g., in Bauer, Furutani, Iwasaki [6]; see also Calin, Chang, Furutani, Iwasaki [32,Sect. 10.3] for the Heat kernel; more generally, see Beals, Gaveau, Greiner, Kannai [15] for operators lifting to sub-Laplacians on 2-step Carnot groups).…”

Section: Introductionmentioning

confidence: 99%

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The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

Biagi

Bonfiglioli

2017

Proc. London Math. Soc.

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Abstract. We prove the existence of a global fundamental solution Γ(x; y) (with pole x) for any Hörmander operator L = m i=1 X 2 i on R n which is δ λ -homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps δ λ of the form δ λ (x) = (λ σ 1 x 1 , . . . , λ σn xn), with 1 = σ 1 ≤ · · · ≤ σn. Due to a global lifting method for homogeneous operators proved by Folland in [On the Rothschild-Stein lifting theorem, Comm. PDEs, 1977], there exists a Carnot group G and a polynomial surjective map π : G → R n such that L is π-related to a sub-Laplacian L G on G. We show that it is always possible to perform a (global) change of variable on G such that the lifting map π becomes the projection of G ≡ R n ×R p onto R n . We prove that an integration argument over the (non-compact) fibers of π provides a fundamental solution for L. Indeed, if Γ G (x, x ′ ; y, y ′ ) (x, x ′ ∈ R n ; y, y ′ ∈ R p ) is the fundamental solution of L G , we show that Γ G (x, 0; y, y ′ ) is always integrable w.r.t. y ′ ∈ R p , and its y ′ -integral is a fundamental solution for L.

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“…Here λ is a complex parameter located in the strip | Re(λ)| < N + k − 1. As is pointed out in [2] P k,0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1-step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian Δ λ which has been widely studied before in the framework of pseudo-convex domains and CR geometry.…”

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confidence: 98%

The inverse of a parameter family of degenerate operators and applications to the Kohn-Laplacian

Bauer

Furutani

Iwasaki

2015

Advances in Mathematics

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By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family P k,λ of degenerate second order differential operators on R N + . Here λ is a complex parameter located in the strip | Re(λ)| < N + k − 1. As is pointed out in [2] P k,0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1-step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian Δ λ which has been widely studied before in the framework of pseudo-convex domains and CR geometry. As an application we show that in all cases the fundamental solutions have a meromorphic extension in the parameter λ to C \ Q. All poles are simple and Q ⊂ R is an explicitly given discrete set. We recover the invertibility of Δ 1 modulo * Corresponding author.

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Fundamental solution of a higher step Grushin type operator

Cited by 25 publications

References 18 publications

Spectral zeta function on pseudo H-type nilmanifolds

Spectral zeta function on pseudo H-type nilmanifolds

The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

The inverse of a parameter family of degenerate operators and applications to the Kohn-Laplacian

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