PALEY PROJECTIONS ON ANISOTROPIC SOBOLEV SPACES ON TORI

An anisotropic Sobolev space L(S)1(T(d)) on the d-dimensional torus T(d) has an invariant complemented subspace isomorphic to an infinite-dimensional Hilbert space it and only if either the smoothness S (a finite subset of R(d) consisting of points with integer-valued non-negative coordinates and containing the origin) contains two points, one corresponding to a partial derivative of even order and the second to a partial derivative of odd order, and there exists a hyperplane passing through these points which supports the convex hull of S and is not parallel to any axis of R(d). or the same property has one of the lower-dimensional smoothnesses being the intersection of S with some number of coordinate hyperplanes. The simplest example of this condition being satisfied is the 2-dimensional smoothness generated by the points corresponding to the partial derivatives D(x) and D(yy).