Lipschitz $\left(\mathfrak{m}^L\left(s;q\right),p\right)$ and $\left(p,\mathfrak{m}^L\left(s;q\right)\right)-$summing maps

Abstract

Building upon the linear version of mixed summable sequences in arbitrary Banach spaces of A. Pietsch, we introduce a nonlinear version of his concept and study its properties. Extending previous work of J. D. Farmer, W. B. Johnson and J. A. Ch\'avez-Dom\'inguez, we define Lipschitz $\left(\mathfrak{m}^L\left(s;q\right),p\right)$ and Lipschitz $\left(p,\mathfrak{m}^L\left(s;q\right)\right)-$summing maps and establish inclusion theorems, composition theorems and several characterizations. Furthermore, we prove that the classes of Lipschitz $\left(r,\mathfrak{m}^L\left(r;r\right)\right)-$summing maps with $0<r<1$ coincide. We obtain that every Lipschitz map is Lipschitz $\left(p,\mathfrak{m}^L\left(s;q\right)\right)-$summing map with $1\leq s< p$ and $0<q\leq s$ and discuss a sufficient condition for a Lipschitz composition formula as in the linear case of A. Pietsch. Moreover, we discuss a counterexample of the nonlinear composition formula, thus solving a problem by J. D. Farmer and W. B. Johnson.