LAUSR

A famous result of S. Kwapien asserts that a linear operator from a Banach space to a Hilbert space is absolutely $1$-summing whenever its adjoint is absolutely $q$-summing for some $1\leq q<\infty$; this result was recently extended to Lipschitz operators by Chen and Zheng. In the present paper we show that Kwapien's and Chen--Zheng theorems hold in a very relaxed nonlinear environment, under weaker hypotheses. Even when restricted to the original linear case, our result generalizes Kwapien's theorem because it holds when the adjoint is just almost summing. In addition, a variant for $\mathcal{L}_{p}$-spaces, with $p\geq2$, instead of Hilbert spaces is provided.