John and Uniform Domains in Generalized Siegel Boundaries
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John and Uniform Domains in Generalized Siegel Boundaries Roberto Monti1 · Daniele Morbidelli2 Received: 17 May 2018 / Accepted: 7 July 2019 / © Springer Nature B.V. 2019
Abstract Given the pair of vector fields X = ∂x + |z|2m y∂t and Y = ∂y − |z|2m x∂t ,where (x, y, t) = , we give a condition on a bounded domain which ensures that is an (ε, δ)-domain for the Carnot-Carath´eodory metric. We also analyze the Ahlfors regularity of the natural surface measure induced on ∂ by the vector fields. Keywords SubRiemannian distance · John domains · (ε, δ) domains Mathematics Subject Classification (2010) Primary 53C17; Secondary 49J15
1 Introduction In R3 = C × R we consider the vector fields X = ∂x + |z|2m y∂t
and
Y = ∂y − |z|2m x∂t ,
(1.1)
where (x, y, t) = (z, t) ∈ R3 = C × R and m ∈ [1, +∞[ is a real parameter. The vector fields X and Y naturally arise as the real and imaginary part of the holomorphic vector field tangent to the boundary of the generalized Siegel domain {(z1 , z2 ) ∈ C2 : Im z2 > 1 2m+2 }. 2m+2 |z1 | We study the interaction of the Carnot-Carath´eodory (CC) distance d induced by X and Y with the geometry of a surface embedded in R3 . Namely, we give conditions on the boundary ∂ such that an open set ⊂ R3 is a John domain, a uniform domain and such that the natural surface measure induced on ∂ by X and Y is Ahlfors regular, see Definition 1.4. John domains are also known as domains with the twisted cone property, see Definition 5.1. When the distance is induced by H¨ormander vector fields in Rn , several authors proved Daniele Morbidelli
[email protected] Roberto Monti [email protected] 1
Dipartimento di Matematica “Tullio Levi-Civita”, Universit`a degli Studi di Padova, Padova, Italy
2
Dipartimento di Matematica, Universit`a di Bologna, Bologna, Italy
R. Monti, D. Morbidelli
that a bounded John domain supports a global Sobolev-Poincar´e inequality, see [4, 5, 12, 22] and the discussion for a general metric space in [11]. The exterior twisted cone property is also relevant in classical potential theory because it implies the subelliptic Wiener criterion (see [19]). Uniform domains are also known as (ε, δ)-domains, see Definition 6.1. They form a subset of John domains. In the global theory of Sobolev spaces for H¨ormander vector fields, Garofalo and Nhieu proved in [6] that subelliptic Sobolev functions in a uniform domain can be extended to the whole space. In [3] it is also shown that the trace of a Sobolev function in a uniform domain with Ahlfors regular boundary belongs to a suitable Besov space of the boundary. Also for this reason, we shall study the Ahlfors property very carefully. The trace problem was analyzed in [13] in the non-characteristic case and in a two-dimensional model. In [7], we study by a direct approach the trace problem at the boundary of the characteristic half plane t > 0 for vector fields of Martinet type X = ∂x , Y = ∂y + |x|α ∂t in R3 . In spite of the previous results, there are not many examples of John and uniform domains in Carnot-Carath´e
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