There is no local unconditional structure in anisotropic spaces of smooth functions

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

“…The first assertion of this lemma is obvious, and the formula may be verified by simple calculations (by the way, its analog in the two-dimensional setting was proved in [13]).…”

“…It is well-known that for n ≥ 2 the space C k (T n ) of k times continuously differentiable functions on the torus T n is not isomorphic to the space C(T n ). This fact was first announced in [4] and later generalized in many directions (see [5,7,8,14,19,16,13,9,10,15]). However, the most general and natural framework was introduced only in the quite recent paper [11] (see also the preprint [12] for the two-dimensional case).…”

“…At this moment, we will apply some facts from Banach space theory. The procedure is essentially the same as in [13].…”

“…In [13] the following statements were proved in the case when all T j are differential monomials. If, again, there are at least two linearly independent operators among the senior parts of T j (since each T j is a differential monomial, its senior part is either T j or zero), then the space C T (T n ) does not have local unconditional structure.…”

“…The system of equations in Fact 1 is solvable iff the following compatibility condition holds (see [11]): l j=0 ξ j 1 ξ −jκ 2 h j (ξ) = 0 for ξ = 0. We will prove Theorem 1 mainly by combining the techniques from [13] and [11].…”