Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups

Abstract

Abstract. The L 1 -Sobolev inequality states that for compactly supported functions u on the Euclidean n-space, the L n/(n−1) -norm of a compactly supported function is controlled by the L 1 -norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n−1) -norm of a compactly supported differential h-form is controlled by the L 1 -norm of its exterior differential du and its exterior codifferential δu (in special cases the L 1 -norm must be replaced by the H 1 -Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.

Poincaré and Sobolev Inequalities for Differential Forms in Heisenberg Groups and Contact Manifolds

“…For the case p = 1, global inequalities in R n (Gagliardo-Nirenberg inequalities for differential forms) have been proved by Bourgain & Brezis [15] and Lanzani & Stein [36] via a suitable identity for closed differential forms and relying on careful estimates for divergence-free vector fields. Thanks to the counterpart of this identity proved by Chanillo & van Schaftingen in homogeneous groups [18], similar global inequalities for differential forms in H n were proved in [3]. We stress that in [3], algebra plays an important role precisely in the proof of the identities for closed forms.…”

“…Thanks to the counterpart of this identity proved by Chanillo & van Schaftingen in homogeneous groups [18], similar global inequalities for differential forms in H n were proved in [3]. We stress that in [3], algebra plays an important role precisely in the proof of the identities for closed forms. Therefore apart from Heisenberg groups, only a handful of more general nilpotent groups have been treated [11].…”

“…On the contrary, interior inequalities require a different more sophisticated argument (see § 1.7 for a gist of our proof). At the same time, the techniques introduced in the present paper differ substantially from those of [3] for global inequalities for p = 1.…”

“…Interior inequalities when p = 1 use the estimate of [3] combined with an approximate homotopy formula introduced in the present paper, but require a new different argument to control the commutator between Rumin's exterior differential (or de Rham's exterior Poincaré and Sobolev inequalities for differential forms 7 differential in R n ) and multiplication by a cut-off function. These inequalities are proved for Heisenberg groups in [6] and in [4] for Euclidean spaces.…”

“…12 (See[3, Theorem 4.6]). If 0 h 2n + 1, denote by a the order of H,h with respect to group dilations (by Remark 3.10, a = 2 if h = n, n + 1 and a = 4 if h = n, n + 1).…”