ON THE KURATOWSKI MEASURE OF NONCOMPACTNESS FOR DUALITY MAPPINGS

Let (X, parallel to.parallel to) be an infinite dimensional real Banach space having a Frechet differentiable norm and phi: R+ -> R+ be a gauge function. Denote by J(phi): X -> X* the duality mapping on X corresponding to phi. Then, for the Kuratowski measure of noncompactness of J(phi), the following estimate holds: alpha(J(phi)) >= sup {phi(r)/r vertical bar r > 0}. In particular, for -Delta(p): W-0(1,P)(Omega) -> W--1,W-P'(Omega), 1 < p < infinity, 1/p +1/p' = 1, viewed as duality mapping on W-0(1,p)(Omega), corresponding to the gauge function phi(t) = t(p-1), one has alpha(-Delta(p)) ={1 for p = 2, infinity for p is an element of (1,2) boolean OR (2, infinity).