Differential expressions with mixed homogeneity and spaces of smooth functions they generate

Abstract

Let $T_1,...,T_l$ 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 $T_{j}f$, $j=1,...,l$, 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) $ta{u}_1,...,ta{u}_l$ from the initial operators $T_1,...,T_l$. Let $N$ be the dimension of the linear span of ${\tau }_1,...,{\tau }_l$. If $N⩾2$, 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.

arXiv comment31 pages