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Trace theorem on the Heisenberg group. (English) Zbl 1176.46037
The Sobolev space $$H^s(\mathbb{H}^d)$$ on the Heisenberg group is defined for an integer $$s\geq 0$$ by
$H^s(\mathbb{H}^d)= \{f\in L^2(\mathbb{R}^{2d+1}_{x,y,t}),\;JZ^\alpha f\in L^2(\mathbb{R}^{2d+1}_{x,y,t})\text{ for }|\alpha|\leq s\},$ where $$Z_j= D_{x_j}+ y_j D_t$$ for $$j= 1,\dots, d$$, and $$Z_j= D_{y_j}- x_j D_t$$ for $$j= d+1,\dots, 2d$$. The definition extends by interpolation to any real $$s\geq 0$$.
The authors study the trace map $$f\in H^1(\mathbb{H}^d)\to f|_\Sigma$$, where $$\Sigma$$ is a given hypersurface in $$\mathbb{H}^d$$. The properties of the trace map depend on the geometry of $$\Sigma$$ with respect to the vector fields $$Z_j$$, $$j= 1,\dots, 2d$$. The case of a non-characteristic hypersurface $$\Sigma$$ was considered already by the same authors in [J. Inst. Math. Jussieu 4, No. 4, 509–552 (2005; Zbl 1089.35016)]. Here, characteristic hypersurfaces are considered. A space $$H^{1/2}(\Sigma)$$ is defined, in a suitable way, so that the trace is a continuous map onto $$H^{1/2}(\Sigma)$$.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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