Some characterizing conditions for the Hardy inequality

In this paper, a new scale of necessary and sufficient conditions for the validity of the Hardy and reverse Hardy inequalities in the cases
$$ 0<\frac{q}{p}<1,\qquad p\in(-\infty,0)\cup(1,\infty), $$
and
$$ 0<\frac{p}{q}<1,\qquad q\in(0,1) $$
is found and estimates for the best constants are derived.