Abstract
We prove Poincaré and Logβ-Sobolev inequalities for a class of probability measures on step-two Carnot groups.
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Appendix: Proof of Lemma 2
Appendix: Proof of Lemma 2
Proof
We first compute ∇N = (XiN)i= 1,...,n.
Therefore,
where we used that for each skew-symmetric matrix Λ(k), all k ∈{1,...,m}, we have that
From (1), that with some constants \(A,C\in (0,\infty ),\) we have
By choosing \(a\in (0,\infty )\) sufficiently small, we can ensure that C ≤ 1. We note that using antisymmetry of matrices \({\Lambda }_{il}^{\left (k\right )}\) we get
Next, we compute
Hence, we obtain
which after simplifications yields
Thus, we get
which can be represented as follows
Hence, there exists a constant \(B\in (0,\infty )\) such that
Remark 1
If a > 0 is small, ΔN ≥ 0. For large a, in some directions ΔN can be negative. □
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Bou Dagher, E., Zegarliński, B. Coercive Inequalities and U-Bounds on Step-Two Carnot Groups. Potential Anal 59, 589–612 (2023). https://doi.org/10.1007/s11118-021-09979-0
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DOI: https://doi.org/10.1007/s11118-021-09979-0
Keywords
- Poincaré inequality
- Logarithmic-Sobolev inequality
- Carnot groups
- Sub-gradient
- Probability measures
- Kaplan norm