Details
| Original language | English |
|---|---|
| Article number | 107186 |
| Journal | Advances in mathematics |
| Volume | 369 |
| Early online date | 5 May 2020 |
| Publication status | Published - 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.
Keywords
- Bessel functions, Distributions, Homogeneous Lie groups, Left invariant homogeneous differential operator, Local solvability
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Advances in mathematics, Vol. 369, 107186, 08.2020.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Fundamental solutions of a class of ultra-hyperbolic operators on pseudo H-type groups
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?.
PY - 2020/8
Y1 - 2020/8
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.
AB - 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.
KW - Bessel functions
KW - Distributions
KW - Homogeneous Lie groups
KW - Left invariant homogeneous differential operator
KW - Local solvability
UR - http://www.scopus.com/inward/record.url?scp=85084215407&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2020.107186
DO - 10.1016/j.aim.2020.107186
M3 - Article
AN - SCOPUS:85084215407
VL - 369
JO - Advances in mathematics
JF - Advances in mathematics
SN - 0001-8708
M1 - 107186
ER -