On the Uniform Convexity and Uniform Smoothness of Infinite-Order Sobolev Spaces

摘要

Infinite-order Sobolev spaces arise in consideration of nonlinear differentia] equations of arbitrary order. The study of these spaces was initiated by Dubinskii [1, 2], In [3, 4], this author studied properties of infinite-order Sobolev spaces from the point of view of the theory of nuclear Frechet spaces. In particular, it was proved that such spaces are superreflexive under certain conditions. Results of Enflo and Pisier [5] imply that the superreflexivity of a Banach space is equivalent to the existence of a uniformly convex norm with convexity modulus of power growth. Beauzamy [6] suggested a fairly general method for constructing uniformly convex norms on superreflexive Banach spaces.