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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-Banachspaces. Note that embedding theorem between weight variable Lebesgue spaces isproved. In particular, we show that $L_{p(x), \omega}(\Omega)$ for $0< p(x)< 1$isn't locally convex. Also, in this paper a some two-weight estimates for Hardyoperator are proved.

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