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In this paper we obtain two-weight Hardy inequalities on general metricmeasure spaces possessing polar decompositions. Moreover, we also findnecessary and sufficient conditions for the weights for such inequalities to betrue. As a consequence, we establish Hardy, Hardy-Sobolev,Hardy-Littlewood-Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberginequalities and their critical versions on general connected Lie groups, whichinclude both unimodular and non-unimodular cases in compact and noncompactsettings. As a byproduct, it also gives, as a special case, an alternativeproof for Sobolev embedding theorems on general (non-unimodular) Lie groups. Wealso obtain the corresponding uncertainty type principles.

Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups