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Consider a finite collection $\{T_1, \ldots, T_J\}$ of differential operatorswith constant coefficients on $\mathbb{T}^n$ ($n\geq 2$) and the space ofsmooth 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 ifthere are at least two linearly independent operators among their senior parts(relative to some mixed pattern of homogeneity), then this space does not havelocal unconditional structure. This fact generalizes the previously knownresult that such spaces are not isomorphic to a complemented subspace of $C(S)$

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