Weighted Hardy-type inequalities on the cone of quasi-concave functions

The paper is devoted to the study of weighted Hardy-type inequalities on the cone of quasi-concave functions, which is equivalent to the study of the boundedness of the Hardy operator between the Lorentz $\Gamma$-spaces. For described inequalities we obtain necessary and sufficient conditions to hold for parameters $q\geqslant1$, $p>0$ and sufficient conditions for the rest of the range of parameters.