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

Abstract

We prove Poincaré and Logβ-Sobolev inequalities for a class of probability measures on step-two Carnot groups.

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

“…Next, for a wide class of Carnot groups of arbitrary step, we will show that condition (11) is satisfied if 𝑁 is as in (23) or, more generally, as in (24). This, once again, will imply the validity of global Poincaré inequalities, and, as before, of the spectral gaps for operator as in (6). 1 To be precise, the quasi-norms 𝑁 𝑗 above are…”

“…The control over the potential that is present in the 𝑈-bounds is granted by the lower bound of the length of the sub-gradient of the quasi-norm. In general it is the control over the potential 𝑈 that is needed for such types of inequalities; see also [Corollary 4.2.4 [27]] where it was proved that, as in the Euclidean setting, a Schrödinger operator with potential 𝑉 as in (6) has discrete spectrum if 𝑉 grows to infinity in all directions.…”

“…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…”

“…Remark 7. We want to point out that the choice of the homogeneous quas-norm 𝑁 on the Carnot group provides a radical difference on the spectrum of the operators of the form (6). To be more precise, for a probability measure on an 𝐻-type group of the form 𝑍 −1 𝑒 −𝑎𝑁 𝑝 , 𝑝 ∈ (1, 2), the corresponding operator (6) has empty essential spectrum when 𝑁 is a Kaplan norm, and does not even have a spectral gap when 𝑁 is the Carnot-Carathéodory distance; see Remark 4.5.4 in [27].…”