关于双线性拟微分算子的Hörmander类

超越Calderón-Zygmund理论的多线性伪微分算子

Multilinear pseudodifferential operators beyond Calderón-Zygmund theory

我们考虑两类多线性拟微分算子。首先，我们证明了符号仅在空间变量上可测的多线性拟微分算子在加权Lebesgue空间中的有界性。这些结果推广了作者关于线性伪伪微分算子的早期工作。其次，我们研究了符号在Hörmander $S^{m}_{ρ, δ}$类中的双线性伪微分算子的有界性.这些结果在$ρ< 1$情形下是新的，即超出了多线性Calderón-Zygmund理论的范围.

一类双线性乘子的Sobolev模估计

Sobolev norm estimates for a class of bilinear multipliers

我们考虑了Demeter和Thiele [9]在二维双线性希尔伯特变换的分类中作为一个特殊情况出现的双线性乘子。本文研究了它们在索伯列夫空间上的有界性。此外，我们研究了结构相似的运营商的符号，也取决于空间变量。新的结果建立在作者在[5]和[10]中研究的仿积型算子的现有L^p估计的基础上。我们的主要目的是强调与Coifman-Meyer乘子和0阶双线性伪微分算子的类比。

Bilinear Fourier multipliers and the rate of decay of their derivatives

We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain rate of decay of the symbol itself while theorems of the second type require, in addition, the same rate of decay of all derivatives of the symbol. We show that even though these two types of bilinear multiplier theorems are closely related, there are some fundamental differences between them which arise in limiting cases. Also, since theorems of the latter ty…

XMO and Weighted Compact Bilinear Commutators

To study the compactness of bilinear commutators of certain bilinear Calderón--Zygmund operators which include (inhomogeneous) Coifman--Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue [Rev. Mat. Iberoam. 36 (2020), 939--956] introduced a new subspace of BMO$\,(\mathbb{R}^n)$, denoted by XMO$\,(\mathbb{R}^n)$, and conjectured that it is just the space VMO$\,(\mathbb{R}^n)$ introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO$\,(\mathb…