On Hardy-type inequalities in weighted variable exponent spaces $L_{p(x),\omega}$ for $0<p(x)<1$

In this paper two-weighted inequalities for the Hardy operator and its dual operator acting from one weighted variable Lebesgue space to another weighted variable Lebesgue space are proved. In particular, sufficient conditions on the weights ensuring the validity of two-weighted inequalities of Hardy type are found. Also an embedding theorem for weighted variable Lebesgue spaces is proved.