Let $1<p\leq q<\infty .$ Inspired by some recent results concerning Hardy type inequalities we state and prove directly the equivalence of four scales of integral conditions. By applying our result to the original Hardy type inequality situation we obtain a new proof of a number of characterizations of the Hardy inequality and obtain also some new weight characterizations. As another application we prove some new weight characterizations for embeddings between some Lorentz spaces.
"An equivalence theorem for integral conditions related to Hardy's inequality.." Real Anal. Exchange 29 (2) 867 - 880, 2003-2004.