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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 \(H^{1/2}(\Sigma)\).

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI Numdam EuDML
[1] Bahouri, Hajer; Chemin, Jean-Yves; Gallagher, Isabelle, Refined Hardy inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5, 3, 375-391, (2006) · Zbl 1121.43006
[2] Bahouri, Hajer; Chemin, Jean-Yves; Xu, Chao-Jiang, Trace and trace lifting theorems in weighted Sobolev spaces, J. Inst. Math. Jussieu, 4, 4, 509-552, (2005) · Zbl 1089.35016
[3] Bahouri, Hajer; Chemin, Jean-Yves; Xu, Chao-Jiang, Phase space analysis of partial differential equations, 69, Trace theorem on the Heisenberg group on homogeneous hypersurfaces, 1-15, (2006), Birkhäuser Boston, Boston, MA · Zbl 1127.35322
[4] Bahouri, Hajer; Gérard, Patrick; Xu, Chao-Jiang, Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg, J. Anal. Math., 82, 93-118, (2000) · Zbl 0965.22010
[5] Bergh, Jöran; Löfström, Jörgen, Interpolation spaces. An introduction, (1976), Springer-Verlag, Berlin · Zbl 0344.46071
[6] Berhanu, S.; Pesenson, I., The trace problem for vector fields satisfying Hörmander’s condition, Math. Z., 231, 1, 103-122, (1999) · Zbl 0924.46026
[7] Bony, Jean-Michel; Chemin, Jean-Yves, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, 122, 1, 77-118, (1994) · Zbl 0798.35172
[8] Cancelier, C. E.; Chemin, J.-Y.; Xu, C. J., Calcul de Weyl et opérateurs sous-elliptiques, Ann. Inst. Fourier (Grenoble), 43, 4, 1157-1178, (1993) · Zbl 0797.35008
[9] Chemin, Jean-Yves; Xu, Chao-Jian, Inclusions de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques, Ann. Sci. École Norm. Sup. (4), 30, 6, 719-751, (1997) · Zbl 0892.35161
[10] Danielli, Donatella; Garofalo, Nicola; Nhieu, Duy-Minh, Trace inequalities for Carnot-Carathéodory spaces and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27, 2, 195-252 (1999), (1998) · Zbl 0938.46036
[11] Huet, Denise, Décomposition spectrale et opérateurs, (1976), Presses Universitaires de France, Paris · Zbl 0334.47015
[12] Pesenson, I., The trace problem and Hardy operator for nonisotropic function spaces on the Heisenberg group, Comm. Partial Differential Equations, 19, 3-4, 655-676, (1994) · Zbl 0813.31005
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