149 research outputs found
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights
This is the fourth article of our series. Here, we study weighted norm
inequalities for the Riesz transform of the Laplace-Beltrami operator on
Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the
doubling volume property and Gaussian upper bounds.Comment: 12 pages. Fourth of 4 papers. Important revision: improvement of main
result by eliminating use of Poincar\'e inequalities replaced by the weaker
Gaussian keat kernel bound
On isoperimetric profiles of product spaces
Let p ∈ [1,+∞]. Given the Lp-isoperimetric profile of two non-compact Riemannian manifolds M and N, we compute the Lp-isoperimetric profile of the product M×N
Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities
We prove a family of uncertainty inequalities on fairly general groups
and homogeneous spaces, both in the smooth and in the discrete setting. The
crucial point is the proof of the endpoint, which is derived from a
general weak isoperimetric inequality.Comment: 17 page
Spectral density and Sobolev inequalities for pure and mixed states
We prove some general Sobolev-type and related inequalities for positive
operators A of given ultracontractive spectral decay, without assuming e^{-tA}
is submarkovian. These inequalities hold on functions, or pure states, as
usual, but also on mixed states, or density operators in the quantum mechanical
sense. This provides universal bounds of Faber-Krahn type on domains, that
apply to their whole Dirichlet spectrum distribution, not only the first
eigenvalue. Another application is given to relate the Novikov-Shubin numbers
of coverings of finite simplicial complexes to the vanishing of the torsion of
some l^{p,2}-cohomology
Generalized uncertainty inequalities
In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are
obtained for a pair of positive-self adjoint operators on a Hilbert space,
whose spectral projectors satisfy a ``balance condition'' involving certain
operator norms. This result is then applied to obtain uncertainty inequalities
on Riemannian manifolds, Riemannian symmetric spaces of non-compact type,
homogeneous graphs and unimodular Lie groups with sublaplacians.Comment: 19 page
Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups
In the present paper, we develop geometric analytic techniques on Cayley
graphs of finitely generated abelian groups to study the polynomial growth
harmonic functions. We develop a geometric analytic proof of the classical
Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic
functions on lattices \mathds{Z}^n that does not use a representation formula
for harmonic functions. We also calculate the precise dimension of the space of
polynomial growth harmonic functions on finitely generated abelian groups.
While the Cayley graph not only depends on the abelian group, but also on the
choice of a generating set, we find that this dimension depends only on the
group itself.Comment: 15 pages, to appear in Ann. Global Anal. Geo
Towards a unified theory of Sobolev inequalities
We discuss our work on pointwise inequalities for the gradient which are
connected with the isoperimetric profile associated to a given geometry. We
show how they can be used to unify certain aspects of the theory of Sobolev
inequalities. In particular, we discuss our recent papers on fractional order
inequalities, Coulhon type inequalities, transference and dimensionless
inequalities and our forthcoming work on sharp higher order Sobolev
inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1
Compact -deformation and spectral triples
We construct discrete versions of -Minkowski space related to a
certain compactness of the time coordinate. We show that these models fit into
the framework of noncommutative geometry in the sense of spectral triples. The
dynamical system of the underlying discrete groups (which include some
Baumslag--Solitar groups) is heavily used in order to construct \emph{finitely
summable} spectral triples. This allows to bypass an obstruction to
finite-summability appearing when using the common regular representation. The
dimension of these spectral triples is unrelated to the number of coordinates
defining the -deformed Minkowski spaces.Comment: 30 page
Random walks on combs
We develop techniques to obtain rigorous bounds on the behaviour of random
walks on combs. Using these bounds we calculate exactly the spectral dimension
of random combs with infinite teeth at random positions or teeth with random
but finite length. We also calculate exactly the spectral dimension of some
fixed non-translationally invariant combs. We relate the spectral dimension to
the critical exponent of the mass of the two-point function for random walks on
random combs, and compute mean displacements as a function of walk duration. We
prove that the mean first passage time is generally infinite for combs with
anomalous spectral dimension.Comment: 42 pages, 4 figure
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