New characterizations of Sobolev spaces on the Heisenberg group

The main aim of this paper is to present some new characterizations of Sobolev spaces on the Heisenberg group HH. First, among several results (Theorem 1.1 and Theorem 1.2), we prove that if f∈Lp(H)f∈Lp(H), p>1p>1, then f∈W1,p(H)f∈W1,p(H) if and only ifsup0<δ<1⁡∬|f(u)−f(v)|>δδp|u−1v|Q+pdudv<∞. This characterizations is in the spirit of the work by Bourgain, Brezis and Mironescu [5], in particular, the work by Hoai-Minh Nguyen [29] in the Euclidean spaces. Our work extends that of Nguyen to Sobolev spaces W1,p(H)W1,p(H) for p>1p>1 in the setting of Heisenberg group. Second, corresponding to the case p=1p=1, we give a characterizations of BV functions on the Heisenberg group (Theorem 4.1 and Theorem 4.2). Third, we give some more generalized characterizations of Sobolev spaces on the Heisenberg groups (Theorem 5.1 and Theorem 5.2).It is worth to note that the underlying geometry of the Euclidean spaces, such as that any two points in RNRN can be connected by a line-segment, plays an important role in the proof of the main theorems in [29]. Thus, one of the main techniques in [29] is to use the uniformity in every directions of the unit sphere in the Euclidean spaces. More precisely, to deal with the general case σ∈SN−1σ∈SN−1, it is often assumed that σ=eN=(0,…,0,1)σ=eN=(0,…,0,1) and, hence, one just needs to work on the one-dimensional case. This can be done by using the rotation in the Euclidean spaces. Due to the non-commutative nature of the Heisenberg group, the absence of this uniformity on the Heisenberg group creates extra difficulties for us to handle. Hence, we need to find a different approach to establish this characterization.