  In this paper we obtain quite general and definitive forms for
Hardy-Littlewood type inequalities. Moreover, when restricted to the original
particular cases, our approach provides much simpler and straightforward proofs
and we are able to show that in most cases the exponents involved are optimal.
The technique we used is a combination of probabilistic tools and of an
interpolative approach; this former technique is also employed in this paper to
improve the constants for vector-valued Bohnenblust--Hille type inequalities.
