Paley Projections on Anisotropic Sobolev Spaces on Tori

An anisotropic Sobolev spaceL s 1 ( T d ) on the d ‐dimensional torus R d has an invariant complemented subspace isomorphic to an infinite‐dimensional Hilbert space if 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 .