The fundamental solution of a class of ultra-hyperbolic operators on Pseudo H-type groups

2019/01/24 by Wolfram Bauer, Bauer, Wolfram, André Froehly +3
Mathematics · #22E25 #65M80 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1901.08318

openalex publication_date 2019/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Pseudo H-type Lie groups G_r,s of signature (r,s) are defined via a module action of the Clifford algebra Cℓ_r,s on a vector space V ≅ ℝ²ⁿ. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let N_r,s denote the Lie algebra corresponding to G_r,s. A choice of left-invariant vector fields [X₁, …, X_2n] which generate a complement of the center of N_r,s gives rise to a second order operator Δ_r,s:= (X₁²+ … + X_n²)- (X_n+1²+ … + X_2n² ), which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of Δ_r,s in the case r=0, s>0 and study their properties. In the case of r>0 we prove that Δ_r,s admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of Δ_r,s and the existence of a fundamental solution in the space of Schwartz distributions.

Citations

• On the theory of general partial differential operators

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