This is an announcement for the paper "Differential expressions with mixed homogeneity and spaces of smooth functions they generate" by S. V. Kislyakov, D. V. Maksimov, and D. M. Stolyarov.
Abstract: Let ${T_1,\dots,T_l}$ be a collection of differential operators with constant coefficients on the torus $\mathbb{T}^n$. Consider the Banach space $X$ of functions $f$ on the torus for which all functions $T_j f$, $j=1,\dots,l$, are continuous. Extending the previous work of the first two authors, we analyse the embeddability of $X$ into some space $C(K)$ as a complemented subspace. We prove the following. Fix some pattern of mixed homogeneity and extract the senior homogeneous parts (relative to the pattern chosen) ${\tau_1,\dots,\tau_l}$ from the initial operators ${T_1,\dots,T_l}$. Let $N$ be the dimension of the linear span of ${\tau_1,\dots,\tau_l}$. If $N\geqslant 2$, then $X$ is not isomorphic to a complemented subspace of $C(K)$ for any compact space $K$. The main ingredient of the proof of this fact is a new Sobolev-type embedding theorem.
Archive classification: math.FA math.CA
Remarks: 37 pages
Submitted from: dms239@mail.ru
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1209.2078
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