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

Volume 261 / 2021

Anton Tselishchev Studia Mathematica 261 (2021), 207-225 MSC: Primary 46E15. DOI: 10.4064/sm200629-21-12 Published online: 8 September 2021

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, that is, 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 $C(S)$.

Authors

  • Anton TselishchevChebyshev Laboratory
    St. Petersburg State University
    14th Line V.O., 29
    Sankt Petersburg 199178, Russia
    e-mail

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