Around heat decay on forms and relations of nilpotent Lie groups

Abstract

One knows that the large time heat decay exponent on a nilpotent group is given by half the growing rate of the volume of its large balls. This work deals with the similar problem of trying to interpret geometrically the heat decay on (one) forms. We will show how it is (partially) related to the depth of the relations required to define the group. The tools used apply in general on Carnot-Carathéodory manifolds.

Compensated compactness for differential forms in Carnot groups and applications

“…, θ 4 the dual left invariant forms. The following result is proved in [25]: as in Remark 7.4, an orthonormal basis of E 1 0 is given by {θ 1 , θ 2 }; an orthonormal basis of…”

“…• "Intrinsic forms" and the "intrinsic differential" should define a complex that is exact and self-dual under Hodge * -duality. It turns out that such a complex (in fact a sub-complex of the De Rham complex) has been defined and studied by M. Rumin in [25] and [24] ( [23] for contact structures), so that we are provided with a good setting for our theory. For sake of self-consistency of the paper, we present in Section 2 the main features of this complex, that will be denoted by (E * 0 , d c ), where d c :…”

“…In general, the lack of homogeneity of d c can be described through the notion of weight of vector fields and, by duality, of differential forms (see [25]). Elements of the j-th layer of g are said to have (pure) weight w = j; by duality, a 1-form that is dual of a vector field of (pure) weight w = j will be said to have (pure) weight w = j. Vector fields in the direct sum of the first j − 1 layers of g are said to have weight w < j.…”

“…To overcome the difficulties arising from the lack of homogeneity of d c , we rely on an argument introduced in [25] (when dealing with the notion of CCelliptic complex). Let us give a non rigorous sketch of the argument.…”

“…We rely on the CGGP-calculus (see [5] and Appendix A). In such a way, we obtain a "0-order exterior differential"d c , and eventually a "0-order Laplacian" [25] and [5], has both a right and a left parametrix. Thus, we can mimic the proof we have sketched above for the De Rham complex (again, to work in a precise pseudodifferential calculus allows the composition of different operators).…”