论文标题
真实线的仿射组的双重卷积
Dual convolution for the affine group of the real line
论文作者
论文摘要
真实行的仿射组的傅立叶代数具有自然的标识，作为Banach空间，带有微量级运算符的空间在$ l^2（{\ Mathbb r}^\ times，dt/ | t |）$。在本文中，我们研究了微量级运算符的“双重卷积产品”，该产品对应于傅立叶代数中的尖端产物。回答了Eymard和Terp的工作中提出的一个问题，我们提供了此操作的内在描述，该描述不依赖于与傅立叶代数的识别，并为该仿射组的连接组件获得了相似的结果。在这两种情况下，我们都会在相应的BANACH代数上构建显式推导，从而直接验证推导身份，而无需逆傅里叶变换。我们还启动了$ l^p（{\ Mathbb r}^\ times，dt/ | t |）$ $ p \ in（1,2）\ cup（2，\ infty）$的类似班级操作员的类似Banach代数结构的研究。
The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on $L^2({\mathbb R}^\times, dt/ |t|)$. In this paper we study the "dual convolution product" of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on $L^p({\mathbb R}^\times, dt/ |t|)$ for $p\in (1,2)\cup(2,\infty)$.
