VECTOR-VALUED MORREY'S EMBEDDING THEOREM AND HOLDER CONTINUITY IN PARABOLIC PROBLEMS

If I subset of R is an open interval and Omega subset of R(N) an open subset with partial derivative Omega Lipschitz continuous, we show that the space W(1,p)(I, L(q)(Omega))boolean AND L(p)(I, W(1,q)(Omega)) is continuously embedded in C(0), 1/p, - N/q (Omega x I)boolean AND L(infinity) (boolean AND x I) if p, q is an element of (1, infinity) and q > Np'. When p = q, this coincides with Morrey's embedding theorem for W(1,p)(Omega x I). While weaker results have been obtained by various methods, including very technical ones, the proof given here follows that of Morrey's theorem in the scalar case and relies only on widely known properties of the classical Sobolev spaces and of the Bochner integral. This embedding is useful to formulate nonlinear evolution problems as functional equations, but it has other applications. As an example, we derive apparently new space-time Holder continuity properties for u(t) = Au + f, u(., 0) = u(0) when A generates a holomorphic semigroup on L(q)(Omega).