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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:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:
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