Abstract. We study the Grushin operators acting on Rand defined by the formulaWe obtain weighted Plancherel estimates for the considered operators. As a consequence we prove L p spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These results are sharp if d 1 ≥ d 2 . We discuss also an interesting phenomenon for weighted Plancherel estimates for d 1 < d 2 . The described spectral multiplier theorem is the analogue of the result for the sublaplacian on the Heisenberg group obtained by Müller and Stein and by Hebisch.

Abstract. In a recent work by A. Martini and A. Sikora, sharp L p spectral multiplier theorems for the Grushin operators acting on Rx ′′ and defined by the formulaHere we complete the picture by proving sharp results in the case d 1 < d 2 . Our approach exploits L 2 weighted estimates with "extra weights" depending only on the second factor of R d 1 × R d 2 (in contrast with the mentioned work, where the "extra weights" depend on the first factor) and gives a new unified proof of the sharp results without restrictions on the dimensions.

We study the problem of L p -boundedness (1 < p < ∞) of operators of the form m(L 1 , . . . , Ln) for a commuting system of self-adjoint leftinvariant differential operators L 1 , . . . , Ln on a Lie group G of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when G is a homogeneous group and L 1 , . . . , Ln are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

Abstract. Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let L be a homogeneous sub-Laplacian on G. By a theorem due to Christ and to Mauceri and Meda, an operator of the form F (L) is of weak type (1, 1) and bounded on L p (G) for all p ∈ (1, ∞) whenever the multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2. It is known that, for several 2-step groups and sub-Laplacians, the threshold Q/2 in the smoothness condition is not sharp and in many cases it is possible to push it down to d/2. Here we show that, for all 2-step groups and sub-Laplacians, the sharp threshold is strictly less than Q/2, but not less than d/2.

The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L 1 , . . . , L n on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a "weighted subcoercive operator" of ter Elst and Robinson (1998) [52]. The joint spectrum of L 1 , . . . , L n in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L 1 , . . . , L n . Connections with the theory of Gelfand pairs are established in the case L 1 , . . . , L n generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G).