917,297 research outputs found
A Simple Approach to Functional Inequalities for Non-local Dirichlet Forms
Full text linkWith direct and simple proofs, we establish Poincar\'{e} type inequalities
(including Poincar\'{e} inequalities, weak Poincar\'{e} inequalities and super
Poincar\'{e} inequalities), entropy inequalities and Beckner-type inequalities
for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet
forms with general jump kernel, and also work for settings. Our
results yield a new sufficient condition for fractional Poincar\'{e}
inequalities, which were recently studied in \cite{MRS,Gre}. To our knowledge
this is the first result providing entropy inequalities and Beckner-type
inequalities for measures more general than L\'{e}vy measures.Comment: 12 page
Convex Sobolev inequalities and spectral gap
Get PDFThis note is devoted to the proof of convex Sobolev (or generalized
Poincar\'{e}) inequalities which interpolate between spectral gap (or
Poincar\'{e}) inequalities and logarithmic Sobolev inequalities. We extend to
the whole family of convex Sobolev inequalities results which have recently
been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev
inequalities. Under local conditions on the density of the measure with respect
to a reference measure, we prove that spectral gap inequalities imply all
convex Sobolev inequalities with constants which are uniformly bounded in the
limit approaching the logarithmic Sobolev inequalities. We recover the case of
the logarithmic Sobolev inequalities as a special case
Poincar\'e inequality for non euclidean metrics and transportation cost inequalities on
Get PDFIn this paper, we consider Poincar\'e inequalities for non euclidean metrics
on . These inequalities enable us to derive precise dimension
free concentration inequalities for product measures. This technique is
appropriate for a large scope of concentration rate: between exponential and
gaussian and beyond. We give different equivalent functional forms of these
Poincar\'e type inequalities in terms of transportation-cost inequalities and
infimum convolution inequalities. Workable sufficient conditions are given and
a comparison is made with generalized Beckner-Latala-Oleszkiewicz inequalities
Height inequalities and canonical class inequalities
Full text linkAn expository lecture on the analogy between the subjects of the title.
Delivered at the International Conference on Number Theory at the Korea
Institute for Advanced Study in December 1997.Comment: Not for separate publicatio
Towards a unified theory of Sobolev inequalities
Full text linkWe 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
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