Summary: A new type of input-output stability is defined, based on the use of a Sobolev space \(W\); \(W\) is well suited, like the Lebesgue space \(L_2\), to obtain stability characterizations in the time and frequency domains. Moreover, if compared with \(L_2\), \(W\) has additional properties which enable us to establish ``local'' stability results. A local version of the small gain theorem is established in this framework, as well as some consequences of this result, in particular local versions of the passivity theorem and of the circle criterion. The relationship between ``\(W\)-stability'' and asymptotic stability is studied.