Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups

Abstract

Abstract. Let L be a non-negative self-adjoint N × N matrix-valued operator of order a ≤ Q on a Carnot group G. Here Q is the homogeneous dimension of G. The aim of this paper is to investigate the relationship between hypoellipticity and maximal hypoellipticity (i.e. sharp L 2 estimates in appropriate Sobolev spaces), L p -maximal hypoellipticity (i.e. sharp L p estimates in appropriate Sobolev spaces for 1 < p < ∞), and what we call maximal subellipticity of L (which is basically a sharp higher order energy estimate).

L1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

“…Theorem 4.4 (see [7], Theorem 3.1). If 0 ≤ h ≤ 2n + 1, then the differential operator ∆ H,h is homogeneous of degree a with respect to group dilations, where a = 2 if h = n, n + 1 and a = 4 if h = n, n + 1.…”

“…for every compact subset K and every ǫ > 0, there exists a smooth compactly supported function χ on R n such that χ ≥ 1 on K and R n |dχ| n < ǫ, (see [14] Section 4. 7).…”

Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups

“…u L n/(n−1) (R n ) ≤ C du L 1 (R n ) + δu H 1 (R n ) if h = 1; (5) u L n/(n−1) (R n ) ≤ C du H 1 (R n ) + δu L 1 (R n ) if h = n − 1, (6) where d is the exterior differential, and δ (the exterior codifferential) is its formal L 2 -adjoint. Here H 1 (R n ) is the real Hardy space (see e.g.…”

“…In other words, the main result of [22] provides a priori estimates for a div-curl systems with data in L 1 (R n ). We stress that inequalities (5) and (6) fail if we replace the Hardy norm with the L 1 -norm. Indeed (for instance), the inequality (7) u L n/(n−1) (R n ) ≤ C du L 1 (R n ) + δu L 1 (R n )…”

Faraday’s Form and Maxwell’s Equations in the Heisenberg Group

“…Theorem 5.10 ([21],[4] and[10]). Denote by ∆ H 1 ,1 := δ c d c + 1 16 (d c δ c ) 2 Rumin's Laplacian on intrinsic 1-forms in H 1 .…”