Symmetry and inverse-closedness of some $p$-Beurling algebras

Let $(G,d)$ be a metric space with the counting measure $\mu$ satisfying some growth conditions. Let $\omega (x,y)=(1+d(x,y){)}^{\delta }$ for some $0<\delta \le 1$. Let $0<p\le 1$. Let ${\mathcal{A}}_{p\omega }$ be the collection of kernels $K$ on $G\times G$ satisfying $\max{\sup_x\sum_y |K(x,y)|^p\omega(x,y)^p, \sup_y\sum_x |K(x,y)|^p\omega(x,y)^p}<\infty$\mathrm{.}Each$K \in \mathcal A_{p\omega}$definesaboundedlinearoperatoron$\ell^2(G)$. If in addition, $\omega$ satisfies the weak growth condition, then we show that ${\mathcal{A}}_{p\omega }$ is inverse closed in $B({\mathrm{\ell }}^2(G))$. 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.