In this paper we convert a (linear abstract) initial boundary value problem into an abstract Cauchy problem on some product space and use semigroup methods to solve it. In particular, we apply spectral theory in order to discuss stability under boundary feedback. (2000). 47D06, 34G10, 93D15. Mathematics Subject Classification

We prove a sharp multiplier theorem of Mihlin-Hörmander type for the Grushin operator on the unit sphere in R 3 , and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics.2010 Mathematics Subject Classification. 33C55, 42B15 (primary); 53C17, 58J50 (secondary).So the system of vector fields {Z 1 , Z 2 } is 2-step bracket-generating and determines a sub-Riemannian structure on S (more on sub-Riemannian geometry can be found, e.g., in [BeRi, CaCh, Mo]). At each point z ∈ S, the horizontal distributionis given the inner product ·, · z corresponding to the normfor all z ∈ S and v ∈ H z S. Note that the horizontal distribution has not constant rank and degenerates at the equator E = {z ∈ S : z 3 = 0}. If z ∈ E, then dim H z S = 1 and | · | z coincides with (the restriction of) the standard Riemannian norm. If z ∈ S \ E, then dim H z S = 2 and {Z 1 | z , Z 2 | z } is an orthonormal basis of H z S with respect to the inner product ·, · z .

In the spirit of an earlier result of D. Müller on the Heisenberg group we prove a restriction theorem on a certain class of two step nilpotent Lie groups. Our result extends that of Müller also in the framework of the Heisenberg group.

If Q is a real, symmetric and positive definite n × n matrix, and B a real n × n matrix whose eigenvalues have negative real parts, we consider the Ornstein-Uhlenbeck semigroup on R n with covariance Q and drift matrix B. Our main result is that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure. The proof has a geometric gist and hinges on the "forbidden zones method" previously introduced by the third author. For large values of the time parameter, we also prove a refinement of this result, in the spirit of a conjecture due to Talagrand.On the space C b (R n ) of bounded continuous functions, we consider the Ornstein-Uhlenbeck semigroup H t t>0 , explicitly given by Kolmogorov's formula( 1.2)The Gaussian measure γ ∞ is the unique invariant measure of the semigroup H t . We are interested in the maximal operator defined as

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