Absence of Local Unconditional Structure in Spaces of Smooth Functions on Two-Dimensional Torus

Abstract

Consider a finite collection {T1, . . . , TJ} of differential operators with constant coefficients on 𝕋2 and the space of smooth functions generated by this collection, namely, the space of functions f such that Tjf ∈ C(𝕋2). Under a certain natural condition, we prove that this space is not isomorphic to a quotient of a C(S)-space and does not have a local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of C(S).