Hardy’s inequality and its descendants: a probability approach

We formulate and prove a generalization of Hardy’s inequality [27] in terms random variables show that it contains the usual (or familiar) continuous discrete forms inequality. Next we improve recent version by Li Mao [42] with weights for general Borel measures mixed norms so implies Liao [43] Hardy Muckenhoupt [48] as well norm versions due to Littlewood [29], Bliss [8], Bradley [14]. An equivalent formulation is given well. also reverse inequality, closely related Copson Carleman-Pólya-Knopp via variables. Finally connect our counting process martingales survival analysis, briefly discuss other applications.