112 research outputs found
Every P-convex subset of is already strongly P-convex
A classical result of Malgrange says that for a polynomial P and an open
subset of the differential operator is surjective on
if and only if is P-convex. H\"ormander showed that
is surjective as an operator on if and only if
is strongly P-convex. It is well known that the natural question
whether these two notions coincide has to be answered in the negative in
general. However, Tr\`eves conjectured that in the case of d=2 P-convexity and
strong P-convexity are equivalent. A proof of this conjecture is given in this
note
Thermoacoustic tomography with variable sound speed
We study the mathematical model of thermoacoustic tomography in media with a
variable speed for a fixed time interval, greater than the diameter of the
domain. In case of measurements on the whole boundary, we give an explicit
solution in terms of a Neumann series expansion. We give necessary and
sufficient conditions for uniqueness and stability when the measurements are
taken on a part of the boundary
On the Fredholm property of bisingular pseudodifferential operators
For operators belonging either to a class of global bisingular
pseudodifferential operators on or to a class of bisingular
pseudodifferential operators on a product of two closed smooth
manifolds, we show the equivalence of their ellipticity (defined by the
invertibility of certain associated homogeneous principal symbols) and their
Fredholm mapping property in associated scales of Sobolev spaces. We also prove
the spectral invariance of these operator classes and then extend these results
to the even larger classes of Toeplitz type operators.Comment: 21 pages. Expanded sections 3 and 4. Corrected typos. Added
reference
Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds
We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold,
where the L^2 norm of all second derivatives of the function involving
horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian.
Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE
in non-divergence form with bounded measurable coefficients. The method of
proof is through a Bochner technique. The Heisenberg group is seen to be en
extremal manifold for our inequality in the class of manifolds whose Ricci
curvature is non-negative.Comment: 13 page
Spherical Spectral Synthesis and Two-Radius Theorems on Damek-Ricci Spaces
We prove that spherical spectral analysis and synthesis hold in Damek-Ricci
spaces and derive two-radius theorems
Thermoacoustic tomography arising in brain imaging
We study the mathematical model of thermoacoustic and photoacoustic
tomography when the sound speed has a jump across a smooth surface. This models
the change of the sound speed in the skull when trying to image the human
brain. We derive an explicit inversion formula in the form of a convergent
Neumann series under the assumptions that all singularities from the support of
the source reach the boundary
Biinvariant operators on nilpotent Lie groups
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46602/1/222_2005_Article_BF01403051.pd
Homological algebra for Schwartz algebras of reductive p-adic groups
Let G be a reductive group over a non-Archimedean local field. Then the
canonical functor from the derived category of smooth tempered representations
of G to the derived category of all smooth representations of G is fully
faithful. Here we consider representations on bornological vector spaces. As a
consequence, if V and W are two tempered irreducible representations and if V
or W is square-integrable, then Ext_G^n(V,W) vanishes for all n>0. We use this
to prove in full generality a formula for the formal dimension of
square-integrable representations due to Schneider and Stuhler.Comment: 34 pages, version 2 contains, in addition, a discussion about formal
dimensions from the point of view of Schwartz algebras and von Neumann
algebra
Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow
The main result of the paper is Egorov's theorem for transversally elliptic
operators on compact foliated manifolds. This theorem is applied to describe
the noncommutative geodesic flow in noncommutative geometry of Riemannian
foliations.Comment: 23 pages, no figures. Completely revised and improved version of
dg-ga/970301
- …