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Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • Wolfram Bauer
  • André Froehly
  • Irina Markina

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  • University of Bergen (UiB)
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OriginalspracheEnglisch
Aufsatznummer107186
FachzeitschriftAdvances in mathematics
Jahrgang369
Frühes Online-Datum5 Mai 2020
PublikationsstatusVeröffentlicht - Aug. 2020

Abstract

Pseudo H-type Lie groups Gr,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓr,s on a vector space V≅R2n. They form a subclass of all 2-step nilpotent Lie groups. Based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let Nr,s denote the Lie algebra corresponding to Gr,s. In the case s>0 a choice of left-invariant vector fields [X1,…,X2n] which generate a complement of the center of Nr,s gives rise to a second order differential operator Δr,s:=(X1 2+…+Xn 2)−(Xn+1 2+…+X2n 2), which we call ultra-hyperbolic. We prove that Δr,s is locally solvable if and only if r=0. In particular, it follows that Δr,s does not admit a fundamental solution in the space D(Gr,s) of Schwartz distributions whenever r>0. In terms of classical special functions we present families of fundamental solutions of Δ0,s in the class of tempered distributions S(G0,s) and study their properties.

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Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups. / Bauer, Wolfram; Froehly, André; Markina, Irina.
in: Advances in mathematics, Jahrgang 369, 107186, 08.2020.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Bauer W, Froehly A, Markina I. Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups. Advances in mathematics. 2020 Aug;369:107186. Epub 2020 Mai 5. doi: 10.1016/j.aim.2020.107186
Bauer, Wolfram ; Froehly, André ; Markina, Irina. / Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups. in: Advances in mathematics. 2020 ; Jahrgang 369.
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abstract = "Pseudo H-type Lie groups Gr,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓr,s on a vector space V≅R2n. They form a subclass of all 2-step nilpotent Lie groups. Based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let Nr,s denote the Lie algebra corresponding to Gr,s. In the case s>0 a choice of left-invariant vector fields [X1,…,X2n] which generate a complement of the center of Nr,s gives rise to a second order differential operator Δr,s:=(X1 2+…+Xn 2)−(Xn+1 2+…+X2n 2), which we call ultra-hyperbolic. We prove that Δr,s is locally solvable if and only if r=0. In particular, it follows that Δr,s does not admit a fundamental solution in the space D′(Gr,s) of Schwartz distributions whenever r>0. In terms of classical special functions we present families of fundamental solutions of Δ0,s in the class of tempered distributions S′(G0,s) and study their properties.",
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AU - Bauer, Wolfram

AU - Froehly, André

AU - Markina, Irina

N1 - Funding Information: All authors have been supported by the DAAD-NFR project ?Subriemannian structures on Lie groups, differential forms and PDE?; project number (NFR) 267630/F10 and (DAAD) 57344898. The first author acknowledges support through the DFG project BA 3793/6-1 in the framework of the SPP 2026 ?Geometry at Infinity?.

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N2 - Pseudo H-type Lie groups Gr,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓr,s on a vector space V≅R2n. They form a subclass of all 2-step nilpotent Lie groups. Based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let Nr,s denote the Lie algebra corresponding to Gr,s. In the case s>0 a choice of left-invariant vector fields [X1,…,X2n] which generate a complement of the center of Nr,s gives rise to a second order differential operator Δr,s:=(X1 2+…+Xn 2)−(Xn+1 2+…+X2n 2), which we call ultra-hyperbolic. We prove that Δr,s is locally solvable if and only if r=0. In particular, it follows that Δr,s does not admit a fundamental solution in the space D′(Gr,s) of Schwartz distributions whenever r>0. In terms of classical special functions we present families of fundamental solutions of Δ0,s in the class of tempered distributions S′(G0,s) and study their properties.

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KW - Bessel functions

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