2012/04/05 by Pratyoosh Kumar, Kumar, Pratyoosh, Swagato K. Ray +3 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Analysis and Transform Methods #Primary 43A85 #Secondary 22E30 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1204.1127
openalex publication_date 2012/04/05 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
In citeRoe Roe proved that if a doubly-infinite sequence fk of\nfunctions on R satisfies fk+1=(dfk/dx) and |fk(x)|\≤ M for\nall k=0,\± 1,\± 2,... and x\∈ R, then f0(x)=a\sin(x+\φ) where\na and \φ are real constants. This result was extended to Rn by\nStrichartz citeStr where d/dx is substituted by the Laplacian on Rn.\nWhile it is plausible to extend this theorem for other Riemannian manifolds or\nLie groups, Strichartz showed that the result holds true for Heisenberg groups,\nbut fails for hyperbolic 3-space. This negative result can be indeed extended\nto any Riemannian symmetric space of noncompact type. We observe that this\nfailure is rooted in the p-dependance of the Lp-spectrum of the Laplacian\non the hyperbolic spaces. Taking this into account we shall prove that for all\nrank one Riemannian symmetric spaces of noncompact type, or more generally for\nthe harmonic NA groups, the theorem actually holds true when uniform\nboundedness is replaced by uniform "almost Lp boundedness". In addition we\nshall see that for the symmetric spaces this theorem is capable of\ncharacterizing the Poisson transforms of Lp functions on the boundary, which\nsome what resembles the original theorem of Roe on R.\n