2008/02/23 by Susana Coré, Coré, Susana, Daryl Geller +1
Mathematics · #22E25 #22E30 #35S05 #42B15 #42B20 #47G30 #58J40 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #advanced mathematical theories #math.AP #math.FA #msc:22E25 #msc:22E30 #msc:35S05 #msc:42B15 #msc:42B20 #msc:47G30 #msc:58J40
paper · pdf · doi:10.48550/arxiv.0802.3452
29 pages
arxiv created 2008/02/23 · openalex publication_date 2008/02/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We produce, on general homogeneous groups, an analogue of the usual Hörmander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand ``multipliers'' of any given order j, and the operators of convolution with their inverse Fourier transforms, which we here call convolution operators of order j. (Here a ``multiplier'' is an analogue of a Hörmander-type symbol a(x,ξ), which is independent of x.) Specifically, we characterize the space of inverse Fourier transforms of multipliers of any order j, and use this characterization to show that the composition of convolution operators of order j1 and j2 is a convolution operator of order j1+j2.