Abstract. We prove that the multilinear Bohnenblust-Hille is a particular case of a quite general family of optimal inequalities.

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 vectorvalued Bohnenblust-Hille type inequalities.2010 Mathematics Subject Classification. 46G25, 47H60.

The existence of unimodular forms on sequence spaces with small norms is crucial in a variety of problems in modern analysis. As a consequence of our results, we prove that the optimal solution f (n 1 , .

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The main result of the present paper is a new Inclusion Theorem for summing operators, that encompasses several recent similar results as particular cases. As applications, we improve estimates of certain Hardy-Littlewood inequalities for multilinear forms.2010 Mathematics Subject Classification. 46G25, 47H60.