具有连续数据的卷积型算子代数并不总是包含所有秩1算子
Algebras of convolution type operators with continuous data do not always contain all rank one operators
摘要：
设$X(\mathbb{R})$是一个可分的Banach函数空间，使得Hardy-Littlewood极大算子是有界的$X(\mathbb{R})$且在其相伴空间$X'(\mathbb{R})$上. $X(\mathbb{R})$上的连续Fourier乘子代数$C_X(\dot{\mathbb{R}})$定义为$\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$上有界变差的连续函数集关于乘子范数的闭包. C. Fernandes，Yu. Karlovich和第一作者\cite{FKK19}证明，如果空间$X(\mathbb{R})$是自反的，那么紧算子的理想包含在Banach代数$\mathcal{A}_{X(\mathbb{R})}$中，该代数由所有乘法算子$aI$、连续函数$a\in C(\dot{\mathbb{R}})$和所有傅立叶卷积算子$W^0(b)$生成，符号为$b\in C_X(\dot{\mathbb{R}})$。证明了存在可分的非自反Banach函数空间$X(\mathbb{R})$，使得代数$\mathcal{A}_{X(\mathbb{R})}$不包含所有秩为1的算子.特别地，这发生在具有$1<p<\infty$的洛伦兹空间$L^{p,1}(\mathbb{R})$的情况下。
英文摘要：
Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. The algebra $C_X(\dot{\mathbb{R}})$ of continuous Fourier multipliers on $X(\mathbb{R})$ is defined as the closure of the set of continuous functions of bounded variation on $\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author \cite{FKK19} that if the space $X(\mathbb{R})$ is reflexive, then the ideal of compact operators is contained in the Banach algebra $\mathcal{A}_{X(\mathbb{R})}$ generated by all multiplication operators $aI$ by continuous functions $a\in C(\dot{\mathbb{R}})$ and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{\mathbb{R}})$. We show that there are separable and non-reflexive Banach function spaces $X(\mathbb{R})$ such that the algebra $\mathcal{A}_{X(\mathbb{R})}$ does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces $L^{p,1}(\mathbb{R})$ with $1<p<\infty$.
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