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Functional Inequalities in Stratified Lie groups with Sobolev, Besov, Lorentz and Morrey spaces
The study of Sobolev inequalities can be divided in two cases: p = 1 and 1 <
p < +. In the case p = 1 we study here a relaxed version of refined
Sobolev inequalities. When p > 1, using as base space classical Lorentz spaces
associated to a weight from the Arino-Muckenhoupt class Bp, we will study
Gagliardo-Nirenberg inequalities. As a by-product we will also consider
Morrey-Sobolev inequalities. These arguments can be generalized to many
different frameworks, in particular the proofs are given in the setting of
stratified Lie groups
Improved Poincar\'e inequalities
Although the Hardy inequality corresponding to one quadratic singularity,
with optimal constant, does not admit any extremal function, it is well known
that such a potential can be improved, in the sense that a positive term can be
added to the quadratic singularity without violating the inequality, and even a
whole asymptotic expansion can be build, with optimal constants for each term.
This phenomenon has not been much studied for other inequalities. Our purpose
is to prove that it also holds for the gaussian Poincar\'e inequality. The
method is based on a recursion formula, which allows to identify the optimal
constants in the asymptotic expansion, order by order. We also apply the same
strategy to a family of Hardy-Poincar\'e inequalities which interpolate between
Hardy and gaussian Poincar\'e inequalities
The Hardy-Rellich Inequality for Polyharmonic Operators
The Hardy-Rellich inequality given here generalizes a Hardy inequality of
Davies (1984), from the case of the Dirichlet Laplacian of a region
to that of the higher order polyharmonic operators
with Dirichlet boundary conditions. The inequality yields some immediate
spectral information for the polyharmonic operators and also bounds on the
trace of the associated semigroups and resolvents.Comment: 19 pages, 2 diagram
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