Algebras of differential operators on Lie groups and spectral multipliers

Let (X, μ) be a measure space, and let L1, . . . ,Ln be (possibly unbounded) selfadjoint operators on L2(X, μ), which commute strongly pairwise, i.e., which admit a joint spectral resolution E on Rn. A joint functional calculus is then defined via spectral integration: for every Borel function m : Rn → C, m(L) = m(L1, . . . ,Ln) = ∫ Rn m(λ) dE(λ) is a normal operator on L2(X, μ), which is bounded if and only if m - called the joint spectral multiplier associated to m(L) - is (E-essentially) bounded. However, the abstract theory of spectral integrals does not tackle the following problem: to find conditions on the multiplier m ensuring the boundedness of m(L) on Lp(X, μ) for some p ≠ 2. We are interested in this problem when the measure space is a connected Lie group G with a right Haar measure, and L1, . . . ,Ln are left-invariant differential operators on G. In fact, the question has been studied quite extensively in the case of a single operator, namely, a sublaplacian or a higher-order analogue. On the other hand, for multiple operators, only specific classes of groups and specific choices of operators have been considered in the literature. Suppose that L1, . . . ,Ln are formally self-adjoint, left-invariant differential operators on a connected Lie group G, which commute pairwise (as operators on smooth functions). Under the assumption that the algebra generated by L1, . . . ,Ln contains a weighted subcoercive operator --- a notion due to [ER98], including positive elliptic operators, sublaplacians and Rockland operators---we prove that L1, . . . ,Ln are (essentially) self-adjoint and strongly commuting on L2(G). Moreover, we perform an abstract study of such a system of operators, in connection with the algebraic structure and the representation theory of G, similarly as what is done in the literature for the algebras of differential operators associated with Gelfand pairs. Under the additional assumption that G has polynomial volume growth, weighted L1 estimates are obtained for the convolution kernel of the operator m(L) corresponding to a compactly supported multiplier m satisfying some smoothness condition. The order of smoothness which we require on m is related to the degree of polynomial growth of G. Some techniques are presented, which allow, for some specific groups and operators, to lower the smoothness requirement on the multiplier. In the case G is a homogeneous Lie group and L1, . . . ,Ln are homogeneous operators, a multiplier theorem of Mihlin-H\"ormander type is proved, extending the result for a single operator of [Chr91] and [MM90]. Further, a product theory is developed, by considering several homogeneous groups Gj , each of which with its own system of operators; a non-conventional use of transference techniques then yields a multiplier theorem of Marcinkiewicz type, not only on the direct product of the Gj , but also on other (possibly non-homogeneous) groups, containing homomorphic images of the Gj . Consequently, for certain non-nilpotent groups of polynomial growth and for some distinguished sublaplacians, we are able to improve the general result of [Ale94].