We study the functional calculus associated with a hypoelliptic left-invariant differential operator \(\mathcal {L}\) on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator \(\mathcal {L}_0\) on the ‘local’ contraction \(G_0\) of G, as well as of the corresponding Rockland operator \(\mathcal {L}_\infty \) on the ‘global’ contraction \(G_\infty \) of G. We provide asymptotic estimates of the Riesz potentials associated with \(\mathcal {L}\) at 0 and at \(\infty \), as well as of the kernels associated with functions of \(\mathcal {L}\) satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for \(\mathcal {L}\) which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with \(\mathcal {L}\) from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

我们借助相应的罗克兰算子\（\ mathcal {L} _0 \ ）研究与连通和简单连通的幂等李群G上的次椭圆左不变微分算子\（\ mathcal {L} \）相关的函数演算）上的'本地'收缩\（G_0 \）的G ^，以及相应的罗克兰操作者的\（\ mathcal {L} _ \ infty \）上收缩的'全局' \（G_ \ infty \）的ģ。我们提供与\（\ mathcal {L} \）在0和\（\ infty \）相关的Riesz势的渐近估计，以及与满足每个顺序的Mihlin条件的\（\ mathcal {L} \）函数相关的内核。我们还证明了\（\ mathcal {L} \）的一些Mihlin-Hörmander乘法定理，将类似结果推广到非齐次情况。最后，我们将与\（\ mathcal {L} \）关联的'Plancherel测度'的密度的渐近研究从拟齐次亚拉普拉斯算子扩展为偶次幂的拟齐次求和。