论文标题
非均匀算子上的功能性演算
Functional Calculus on Non-Homogeneous Operators on Nilpotent Groups
论文作者
论文摘要
我们研究与相关且简单连接的nilpotent Lie组$ g $在相应\ emph {rockland}操作员$ \ \ \ $ \ mathcal {l} _0 $ g_的$ g_0 $ g_0的$ g_0 $ g_0 $ g_0 $ g_0的$ g_0 $ g_0 $ g_的$ g_0 $ g_0 $ g_0 $ g_0 $ g_0 $ g_ $ \ Mathcal {l} _ \ infty $在$ g $的“全局”收缩$ g_ \ infty $中。 我们提供与$ \ Mathcal {l} $相关的Riesz电位的渐近估计，以$ 0 $和$ \ infty $，以及与$ \ MATHCAL {l} $满足每个订单的Mihlin条件的功能相关的内核。 我们还证明了一些$ \ Mathcal {l} $的Mihlin-Hörmander乘数定理，这些定理将类似的结果推广到非均匀情况。 最后，我们将与$ \ Mathcal {l} $相关的“ Plancherel量度”密度的渐近研究从准同型亚拉平式的情况下扩展到了均匀幂的均匀量。
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 \emph{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.
