Completely bounded Paley projections on anisotropic Sobolev spaces on tori

Abstract: We study the existence of certain completely bounded Paley projection on the anisotropic Sobolev spaces on tori. Our result should be viewed as a generalization of a similar result obtained by Pe{\l}czy{\'n}ski and Wojciechowski. By a transference method, we obtain similar results on the Sobolev spaces on quantum tori.