Lipschitz $p$-summing multilinear operators

We apply the geometric approach provided by $\mathrm{\Sigma }$-operators to develop a theory of $p$-summability for multilinear operators. In this way, we introduce the notion of Lipschitz $p$-summing multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck’s coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pelczy'nsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type tensor norm. Moreover, we introduce the notion of Dunford-Pettis multilinear operators. With them, we characterize when a projective tensor product contains ${\mathrm{\ell }}_1$. Relations between Lipschitz $p$-summing multilinear operators with Dunford-Pettis and Hilbert-Schmidt multilinear operators are given.