By P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi

ISBN-10: 3540075372

ISBN-13: 9783540075370

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**Extra resources for Analyse Harmonique sur les Groupes de Lie**

**Sample text**

Then (i)–(iii) above are immediate. We also observe the identity [Z, Z] = 12 i ∂/∂x3 . In order to obtain a more geometric insight we recall the notion of an embedded CR manifold. Let Ω = {z = (z1 , z2 ) ∈ C2 : φ(z) < 0}, φ ∈ C 2 (C2 ), ∇φ = 0, deﬁne a smooth subset of C2 . The tangent space to ∂Ω at p ∈ ∂Ω is given by Tp ∂Ω = {Z ∈ C2 : Re ∂φ(p), Z = 0}, where ∂φ = ∂ ∂ φ, φ , ∂ z¯1 ∂ z¯2 ¯ 1 +Z2 W ¯ 2 , denotes the complex scalar product. and for Z, W ∈ C2 , Z, W = Z1 W The maximal complex, or horizontal, plane at p is given by Hp ∂Ω = {Z ∈ C2 : ∂φ(p), Z = 0}.

The homogeneous dimension of G is r Q= imi . i=1 Observe that the homogeneous dimension of Hn is 2n + 2. 3. The Haar measure on G coincides with the push-forward of the Lebesgue measure on the Lie algebra g under the exponential map. It is easy to verify that the Jacobian determinant of the dilation δs : G → G is constant, equal to sQ . As with the Heisenberg group, we deﬁne the horizontal gradient of a C 1 function f : G → R by m1 ∇0 f = X1j f X1j . j=1 At various points in this survey we will work in this general setting to emphasize the fact that certain results do not depend on the special structure of H.

46) for every x ∈ M . The analytic form of this condition ﬁrst appeared in the literature in 1967 in the celebrated paper of H¨ormander [149], who proved that it is a suﬃcient condition for hypoellipticity of second order diﬀerential operators of m the form L = i=1 Xj2 . Operators of this form are known as sum of squares or sub-Laplacians. Carnot groups arise naturally as ideal boundaries of noncompact rank 1 symmetric spaces. For instance, Hn is isomorphic to the nilpotent part of the Iwasawa decomposition of U (1, n), the isometry group of the complex hyperbolic space of dimension n.

### Analyse Harmonique sur les Groupes de Lie by P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi

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