Absence of local unconditional structure in spaces of smooth functions on the two-dimensional torus

A. Tselishchev

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 48, Tome 491 (2020), pp. 153-172

Résumé

Consider a finite collection $\{T_1, \ldots, T_J\}$ of differential operators with constant coefficients on $\mathbb{T}^2$ and the space of smooth functions generated by this collection, namely, the space of functions $f$ such that $T_j f \in C(\mathbb{T}^2)$. It is proved that under a certain natural condition this space is not isomorphic to a quotient of a $C(S)$-space and does not have a local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of $C(S)$.

Bibliographie

[1] O. V. Besov, V. P. Il'in, S. M. Nikol'skii, Integral Representation of Functions and Imbedding Theorems, Halsted Press, Washington, 1978 | MR

[2] E. B. Fabes, N. M. Rivière, “Singular integrals with mixed homogeneity”, Stud. Math., 27:1 (1966), 19–38 | DOI | MR | Zbl

[3] Y. Gordon, D. R. Lewis, “Absolutely summing operators and local unconditional structure”, Acta Math., 133 (1974), 27–48 | DOI | MR | Zbl

[4] A. Grothendieck, “Erratum au mémoire: produits tensoriels topologiques et espaces nucléaires”, Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 117–120 | DOI | MR | Zbl

[5] G. M. Khenkin, “Otsutstvie ravnomernogo gomeomorfizma mezhdu prostranstvami gladkikh funktsii ot odnogo i ot $n$ peremennykh ($n\geq 2$)”, Matem. sb., 74:4 (1967), 595–606

[6] S. V. Kislyakov, “Absolyutno summiruyuschie operatory na disk-algebre”, Algebra i analiz, 3:4 (1991), 1–77 | MR

[7] S. V. Kislyakov, “Sobolevskie operatory vlozheniya i neizomorfnost nekotorykh banakhovykh prostranstv”, Funkts. analiz i ego pril., 9:4 (1975), 22–27 | MR

[8] S. V. Kislyakov, There is no local unconditional structure in the space of continuously differentiable functions on the torus, LOMI-R-1-77, L., 1977

[9] S. V. Kislyakov, D. V. Maksimov, “Izomorfnyi tip prostranstva gladkikh funktsii, porozhdennogo konechnym semeistvom differentsialnykh operatorov”, Zap. nauchn. sem. POMI, 327, 2005, 78–97 | Zbl

[10] S. V. Kislyakov, D. V. Maksimov, Izomorfnyi tip prostranstv gladkikh funktsii, porozhdennykh konechnym semeistvom neodnorodnykh differentsialnykh operatorov, Preprint POMI, 6/2009, 2009

[11] S. V. Kislyakov, D. V. Maksimov, D. M. Stolyarov, “Differential expressions with mixed homogeneity and spaces of smooth functions they generate in arbitrary dimension”, J. Funct. Anal., 269 (2015), 3220–3263 | DOI | MR | Zbl

[12] S. V. Kislyakov, D. V. Maksimov, D. M. Stolyarov, Differential expressions with mixed homogeneity and spaces of smooth functions they generate, arXiv: 1209.2078 | MR

[13] S. V. Kislyakov, N. G. Sidorenko, “Absence of a local unconditional structure in anisotropic spaces of smooth functions”, Sibirsk. Mat. Zh., 29:3 (1988), 64–77 | MR

[14] S. Kwapień, A. Pełczyński, “Absolutely summing operators and translation-invariant spaces of functions on compact abelian groups”, Math. Nachr., 94 (1980), 303–340 | DOI | MR | Zbl

[15] D. V. Maksimov, “Izomorfnyi tip prostranstva gladkikh funktsii, porozhdennogo konechnym naborom differentsialnykh operatorov. II”, Zap. nauchn. sem. POMI, 333, 2006, 62–65 | Zbl

[16] A. Pełczyński, K. Senator, “On isomorphisms of anisotropic Sobolev spaces with “classical” Banach spaces and Sobolev-type embedding theorem”, Studia Math., 84, 1986, 169–215 | DOI | MR

[17] A. Pietsch, Operator Ideals, Elsevier, North-Holland, 1980 | MR

[18] N. G. Sidorenko, “Neizomorfnost nekotorykh banakhovykh prostranstv gladkikh funktsii prostranstvu nepreryvnykh funktsii”, Funkts. analiz i ego pril., 21:4 (1987), 91–93 | MR

[19] P. Wojtaszczyk, Banach spaces for analysts, Cambridge University Press, 1991 | MR | Zbl