论文标题
Hörmander类型定理用于多线性伪差异操作员
Hörmander type theorem for multilinear Pseudo-differential operators
论文作者
论文摘要
我们为多线性伪分辨率运算符建立了Hörmander型定理，这也是\ cite {MR4322619}结果的概括{MR4322619}对符号，取决于空间变量。通过假设其符号满足点上的衍生估计值（mihlin-type条件），获得了多线性伪差异操作员的最著名结果，也就是说，它们的符号属于某些符号类$ n $ n $ - $ n $ - $ \ mathcal {s} s}^m_ {p， ρ\ le1 $，$ 0 \leΔ<1 $ for Some $ M \ le 0 $。在本文中，我们将考虑多线性伪分辨率的操作员，其符号的平滑度有限，而不是在功能空间而不是以侧面形式（Hörmander型状态）。我们的符号条件比Mihlin型条件弱，具有两种感觉：一种是，我们仅假设在空间变量和频率变量中的空间变量和低阶导数条件中的一阶导数条件，另一个是我们使用$ l^2 $ -A-A-A-A-A-a-a-a-a-a-a-a-a衍生条件。作为一个应用程序，我们获得了一些与属于类$ n $ - $ - $ \ MATHCAL {s}^{M} {m} _ {ρ，Δ}（ρ，Δ}（\ Mathbb {r}^{r}^{d}^{d}^{d}）$，$ 0 $ρle1 $的符号相关的多线性伪分化属性的映射属性。 0 $。此外，可以指出的是，我们的结果可以应用于不属于传统符号类$ n $ - $ \ mathcal {s}^{m} _ {ρ，δ}（\ mathbb {r}^{d}）$的较宽类别的符号类别。
We establish a Hörmander type theorem for the multilinear pseudo-differential operators, which is also a generalization of the results in \cite{MR4322619} to symbols depending on the spatial variable. Most known results for multilinear pseudo-differential operators were obtained by assuming their symbols satisfy pointwise derivative estimates(Mihlin-type condition), that is, their symbols belong to some symbol classes $n$-$\mathcal{S}^m_{ρ, δ}(\mathbb{R}^d)$, $0 \le δ\le ρ\le1$, $0 \le δ<1$ for some $m \le 0$. In this paper, we shall consider multilinear pseudo-differential operators whose symbols have limited smoothness described in terms of function space and not in a pointwise form(Hörmander type condition). Our conditions for symbols are weaker than the Mihlin-type conditions in two senses: the one is that we only assume the first-order derivative conditions in the spatial variable and lower-order derivative conditions in the frequency variable, and the other is that we make use of $L^2$-average condition rather than pointwise derivative conditions for the symbols. As an application, we obtain some mapping properties for the multilinear pseudo-differential operators associated with symbols belonging to the classes $n$-$\mathcal{S}^{m}_{ρ,δ}(\mathbb{R}^{d})$, $0 \le ρ\le 1$, $0 \le δ<1$, $m \le 0$. Moreover, it can be pointed out that our results can be applied to wider classes of symbols which do not belong to the traditional symbol classes $n$-$\mathcal{S}^{m}_{ρ,δ}(\mathbb{R}^{d})$.
