论文标题
使用连续数据的卷积类型运算符代数并不总是包含所有排名
Algebras of convolution type operators with continuous data do not always contain all rank one operators
论文作者
论文摘要
令$ x（\ mathbb {r}）$为一个可分开的banach功能空间，这样耐心的木材最大运算符的限制为$ x（\ mathbb {r}）$，并且在其助理空间$ x'（\ mathbb {r}）上。连续傅立叶乘数在$ x上的代数$ c_x（\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ dot {\ mathbb {r}}）$ $ \ dot {\ mathbb {r}} = \ mathbb {r} \ cup \ {\ infty \} $相对于乘数标准。 Yu的C. Fernandes证明了这一点。 Karlovich和第一作者\ cite {fkk19}，如果空间$ x（\ mathbb {r}）$具有反射性，那么紧凑型操作员的理想是在Banach代数$ \ Mathcal {a} _} _ {x（x（x（r Mathbb {r} $生成的）中， $ a \ in C（\ dot {\ mathbb {r}}）$，以及所有傅立叶卷积运算符$ w^0（b）$，带有符号$ b \ in c_x（\ dot {\ dot {\ mathbb {r}}}）$。我们表明，有可分离的和非反射的banach函数空间$ x（\ mathbb {r}）$，使得代数$ \ mathcal {a} _ {x（\ mathbb {r}）} $并不包含所有排名一个操作员。特别是，在Lorentz Spaces $ l^{p，1}（\ Mathbb {r}）$的情况下，这发生在$ 1 <p <\ infty $。
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 $1<p<\infty$.
