一类$p$ -Beurling代数的对称性和逆闭性
Symmetry and inverse-closedness of some $p$-Beurling algebras
摘要：
设$(G,d)$是度量空间，其计数测度$μ$满足一定的增长条件。让$ω(x,y)=(1+d(x,y))^δ$为一些$0<δ\leq1$。让PDMATHBMQ。令$\mathcal A_{pω}$是满足$\max\{\sup_x\sum_y |K(x,y)|^pω(x,y)^p, \sup_y\sum_x |K(x,y)|^pω(x,y)^p\}<\infty$的$G\times G$上的内核$K$的集合。每个$K \in \mathcal A_{pω}$在$\ell^2(G)$上定义一个有界线性算子。如果$ω$满足弱增长条件，则证明了$\mathcal A_{pω}$在$B(\ell^2(G))$中是逆闭的.本文还讨论了$\mathbb Z^d$上无限矩阵的$p$ - Banach代数和$\mathbb Z^{2d}$上加权$p$ -可和序列的$p$ - Banach代数的逆闭性。为了证明这些结果，我们证明了$p$ - Banach代数的Hulanicki引理和Barnes引理.
英文摘要：
Let $(G,d)$ be a metric space with the counting measure $μ$ satisfying some growth conditions. Let $ω(x,y)=(1+d(x,y))^δ$ for some $0<δ\leq1$. Let $0<p\leq1$. Let $\mathcal A_{pω}$ be the collection of kernels $K$ on $G\times G$ satisfying $\max\{\sup_x\sum_y |K(x,y)|^pω(x,y)^p, \sup_y\sum_x |K(x,y)|^pω(x,y)^p\}<\infty$. Each $K \in \mathcal A_{pω}$ defines a bounded linear operator on $\ell^2(G)$. If in addition, $ω$ satisfies the weak growth condition, then we show that $\mathcal A_{pω}$ is inverse closed in $B(\ell^2(G))$. We shall also discuss inverse-closedness of $p$-Banach algebra of infinite matrices over $\mathbb Z^d$ and the $p$-Banach algebra of weighted $p$-summable sequences over $\mathbb Z^{2d}$ with the twisted convolution. In order to show these results, we prove Hulanicki's lemma and Barnes' lemma for $p$-Banach algebras.
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