On a weighted variable spaces $L_{p(x),\omega }$ for $0<p(x)<1$ and weighted Hardy inequality

In this paper a weighted variable exponent Lebesgue spaces $L_{p(x),\omega }$ for $0<p(x)<1$ is investigated. We show that this spaces is a quasi-Banach spaces. Note that embedding theorem between weight variable Lebesgue spaces is proved. In particular, we show that $L_{p(x),\omega }(\mathrm{\Omega })$ for $0<p(x)<1$ isn’t locally convex. Also, in this paper a some two-weight estimates for Hardy operator are proved.