1,721,130 research outputs found
Huygens' principle and a Paley-Wiener type Theorem on Damek-Ricci spaces
No full textWe prove that Huygens' principle and the principle of equipartition of energy hold
for the modified wave equation on odd dimensional Damek-Ricci spaces. We also
prove a Paley-Wiener type theorem for the inverse of the Helgason Fourier transform on
Damek-Ricci spaces
The Gelfand transform of homogeneous distributions of Heisenberg type groups
No full textA distribution on a Heisenberg type group of homogeneous dimension Q is a biradial kernel of type a if
it coincides with a biradial function, homogeneous of degree a −Q, and smooth away from the identity.
We prove that a distribution is a biradial kernel of type a, 0
≤a<Q, if and only if its Gelfand transform,
defined on the Heisenberg fan, extends to a smooth even function on the upper half plane, homogeneous
of degree −a/2. A similar result holds for radial kernels on the Heisenberg group
Geodesic inversion and Sobolev spaces on Heisenberg type groups
No full textLet s be the geodesic inversion on a Heisenberg type
group N with homogeneous dimension Q,
and denote by S the jacobian of s.
We prove that,
for -Q/2<a<Q/2, the operators T_a
defined by T_a(f)= S^{1/2-a/Q} (f\circ s)
are bounded on certain homogeneous Sobolev spaces H^a(N)
if and only if N is an Iwasawa N-group
Sobolev spaces and the Cayley transform
No full textThe generalised Cayley transform C
from an Iwasawa N-group
into the corresponding real unit sphere S
induces
isomorphisms between suitable Sobolev spaces
H^\alpha(S) and H^\alpha(N).
We study the differential of C
and we obtain a criterion for a function
to be in H^\alpha(S)
Some properties of horocycles on Damek-Ricci spaces
No full textWe prove that a Damek–Ricci space is symmetric if and only if the geodesic inversion
preserves the set of horocycles
The Cayley transform and uniformly bounded representations
No full textLet G be a simple Lie group of real rank one, with Iwasawa decomposition KA \bar N and Bruhat
big cell NMA\bar N: Then the space G/MA \bar N may be (almost) identified with N and with K /M,
and these identifications induce the (generalised) Cayley transform C : N \to K /M. We show
that C is a conformal map of Carnot–Caratheodory manifolds, and that composition with the
Cayley transform, combined with multiplication by appropriate powers of the Jacobian,
induces isomorphisms of Sobolev spaces on N
and on K/M. We use this to construct
uniformly bounded and slowly growing representations of G
Gelfand transforms of polyradial functions on the Heisenberg group
No full textWe prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups
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