An equivalence theorem for some integral conditions with general measures related to Hardy´s inequality

It is proved that, besides the usual Muckenhoupt condition, there exist four different scales of conditions for characterizing the Hardy type inequality with general measures for the case 1<p<img src="http://www.sciencedirect.com/scidirimg/entities/2a7d.gif" />q<∞. In fact, an even more general equivalence theorem of independent interest is proved and discussed.