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

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 smooth functions generated by this collection, namely, $f$ such that $T_j f \in C(\mathbb{T}^n)$, $1\leq j\leq J$. We prove if there are at least two linearly independent among their senior parts (relative to some mixed pattern homogeneity), then does not have local unconditional structure. This fact generalizes previously known result spaces isomorphic complemented subspace $C(S)$