A NOTE ON UNIQUENESS OF UNIFORM NORM PROPERTY IN THE BEURLING ALGEBRA $L^1(G_1 \times G_2, \omega)$

Let $G_1$ and $G_2$ be LCA groups with identities being $e_1$ and $e_2$, and let $\omega$ be a (Borel measurable) weight function on $G_1 \times G_2$. Let $\overline{\omega}(s, t) = \omega(s, e_2)\omega(e_1, t) \; ((s, t) \in G_1 \times G_2)$. Then $\overline{\omega}$ is also a weight function on $G_1 \times G_2$. In this small note, it is proved that the Beurling algebra $L^1(G_1 \times G_2, \omega)$ has unique uniform norm property iff $L^1(G_1 \times G_2, \overline{\omega})$ has the same property. This result is important because the above statement does not hold true for some properties.