Absence of local unconditional structure in spaces of smooth functions on the torus of arbitrary dimension

TL;DR

This paper proves that certain spaces of smooth functions on the torus, generated by differential operators with specific properties, lack local unconditional structure, extending previous results on their structural limitations.

Contribution

It generalizes known results by showing these function spaces lack local unconditional structure when generated by multiple independent differential operators.

Findings

Spaces lack local unconditional structure under given conditions

Generalizes previous non-isomorphism results

Applicable to spaces generated by differential operators with linearly independent senior parts

Abstract

Consider a finite collection $\{T_1, \ldots, T_J\}$ of differential operators with constant coefficients on $\mathbb{T}^n$ ($n\geq 2$) and the space of smooth functions generated by this collection, namely, the space of functions $f$ such that $T_j f \in C(\mathbb{T}^n)$, $1\leq j\leq J$. We prove that if there are at least two linearly independent operators among their senior parts (relative to some mixed pattern of homogeneity), then this space does not have local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of $C(S)$