Coercive inequalities in higher-dimensional anisotropic heisenberg group

Abstract

In the setting of higher-dimensional anisotropic Heisenberg group, we compute the fundamental solution for the sub-Laplacian, and we prove Poincaré and $$\beta $$ β -Logarithmic Sobolev inequalities for measures as a function of this fundamental solution.

Coercive Inequalities and U-Bounds on Step-Two Carnot Groups

“…indicating also no-go zone of parameters where the corresponding inequality fails. We were able to use the methods of this paper to obtain similar results in the setting of the Higher-Dimensional Anisotropic Heisenberg group [8].…”

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

“…Remark 10 (The anisotropic Heisenberg group). We conclude this part dedicated to step 2 groups by considering a group treated in [8] to which our results apply. The group under consideration is the so called anisotropic Heisenberg group H 2𝑛 1 2 , 1 .…”

“…with respect to a probability measure) 𝑞-Poincaré inequalities. This method was successfully applied in the case of step 2 groups in [26], [27], [6], [28] and [8], and on Carnot groups of higher step in [7]. Note that, by taking…”

“…Let us finally remark that in [8] the authors had to deal with the non trivial problem of finding a fundamental solution of the sub-Laplacian in order to find the suitable probability measure to prove the inequalities. Theorem 6, instead, is direct, and gives already a class of measures for which the inequalities are true.…”

“…Since H 2𝑛 1 2 , 1 is a Carnot group of step 2, we get the validity of global 𝑞-Poincaré inequalities for any smooth homogeneous quasi-norm as in (25) by Theorem 6. However, the validity of such inequalities was first proved in [8] by using probability measures whose density depends on a specific smooth quasi-norm 𝑁, that is, for 𝑁 being the fundamental solution for the sub-Laplacian. The quasi-norm in [8] is…”