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Paley Projections on Anisotropic Sobolev Spaces on Tori
Author(s) -
Pelczynski A.,
Wojciechowski M.
Publication year - 1992
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-65.2.405
Subject(s) - mathematics , sobolev space , hyperplane , torus , combinatorics , pure mathematics , hilbert space , intersection (aeronautics) , invariant (physics) , mathematical analysis , geometry , mathematical physics , engineering , aerospace engineering
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 .