SUPERPOSITION IN HOMOGENEOUS AND VECTOR VALUED SOBOLEV SPACES

We give a sufficient condition on a function f : R-k -> R so that it takes by superposition the homogeneous vector valued space (W) over dot(p)(m) boolean AND (W) over dot(mp)(1) (R-n, R-k) into the corresponding real valued space, for integers m, n, k such that m >= 2, k, n >= 1, and p is an element of [1, +infinity[. In case k = 1, this condition also turns out to be necessary. For k > 1, it is not proved to be necessary, but it is weaker than the conditions used till now, such as the continuity and boundedness of all derivatives up to order m.