An example of a weighted algebra $L_p^w(G)$ on uncountable group

We construct examples of weighted algebras $L_p^w(G)$ with $1<p\le 2$ on uncountable free groups. For $p>2$ no weighted algebras exist on these groups. From the other side, we prove that an amenable group on which exist weighted algebras with $p>1$ must be sigma-compact.