A result by Courrège says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}={\mathcal{L}}^{\sigma ,b}+{\mathcal{L}}^{\mu }$ where${\mathcal{L}}^{\sigma ,b}[u](x)=\mathrm{\text{tr}}(\sigma {\sigma }^{\mathtt{\text{T}}}{D}^{2}u(x))+b\cdot Du(x)$ and${\mathcal{L}}^{\mu }[u](x)=\underset{{\mathbb{R}}^d\setminus \{0\}}{\int }(u(x+z)-u(x)-z\cdot Du(x){\mathbf{1}}_{|z|\le 1})\mathrm{d}\mu (z).$ This class of operators coincides with the infinitesimal generators of Lévy processes in probability theory. In this paper we give a complete characterization of the operators of this form that satisfy the Liouville theorem: Bounded solutions u of $\mathcal{L}[u]=0$ in ${\mathbb{R}}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $\mathcal{L}[u]=0$ in ${\mathbb{R}}^d$. The proofs combine arguments from PDEs and group theory. They are simple and short.

Courrège的结果表明，线性平移不变算符在且仅当其形式为时才满足最大原理 $\mathcal{大号}={\mathcal{大号}}^{\sigma ，b}+{\mathcal{大号}}^{\mu }$ 哪里${\mathcal{大号}}^{\sigma ，b}[ü]（X）=\mathrm{\text{TR}}（\sigma {\sigma }^{\mathtt{\text{Ť}}}{d}^2ü（X））+b\cdot dü（X）$ 和${\mathcal{大号}}^{\mu }[ü]（X）=\underset{{\mathbb{[R}}^d\setminus \{0\}}{\int }（ü（X+ž）-ü（X）-ž\cdot dü（X）{\mathbf{1个}}_{|ž|\le \mathrm{1个}}）\mathrm{d}\mu （ž）。$这类算子与概率论中Lévy过程的无穷小生成器一致。在本文中，我们给出了满足刘维定理这种形式的运营商一个完整的表征：界解ü的$\mathcal{大号}[ü]=0$ 在 ${\mathbb{[R}}^d$是恒定的。Liouville属性是作为周期结果的结果获得的，该结果完全刻画了$\mathcal{大号}[ü]=0$ 在 ${\mathbb{[R}}^d$。证明结合了PDE和组理论的论点。它们简单而简短。