Geometric logarithmic-Hardy and Hardy-Poincaré inequalities on stratified groups
arxiv(2024)
摘要
We develop a unified strategy to obtain the geometric logarithmic Hardy
inequality on any open set M of a stratified group, provided the validity of
the Hardy inequality in this setting, where the so-called "weight" is regarded
to be any measurable non-negative function on M . Provided the legitimacy of
the latter for some open set and for some weight, we also show an inequality
that is an extension of the "generalised Poincaré inequality" introduced by
Beckner with the addition of a weight, and this is referred to as the
"geometric Hardy-Poincaré inequality". The aforesaid inequalities become
explicit in the case where M is the half-space of the group and the weight is
the distance function from the boundary, and in the case where M is just the
whole group (or any open set in the group), in which case the weight is the
"horizontal norm" on the first stratum of the group. For the second case, the
semi-Gaussian analogue of the derived inequalities is proved, when the Gaussian
measure is regarded with respect to the first stratum of the group. Applying
our results to the case where the group is just the (abelian) Euclidean space
we generalise the classical probabilistic Poincaré inequality by adding
weights.
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