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. The embeddability of $X$ into some
space $C(K)$ as a complemented subspace will be discussed. The main result
is as follows. Fix some pattern of mixed homogeneity and extract the
senior homogeneous parts (relative to the pattern chosen)
${\tau_1,...,\tau_l}$ from the initial operators ${T_1,...,T_l}$. If there
are two nonproportional operators among the $\tau_j$ (for at least one
homogeneity pattern), then $X$ is not isomorphic to a complemented
subspace of $C(K)$ for any compact space $K$.
The main ingredient of the proof is a new Sobolev-type embedding
theorem. It generalises the classical embedding of
${\stackrel{\circ}{W}}_1^1(\mathbb{R}^2)$ to $L^2(\mathbb{R}^2)$. The difference is that
now the integrability condition is imposed on certain linear combinations
of derivatives of different order of several functions rather than on the
first order derivatives of one function.
This is a joint work with D. Maksimov and D. Stolyarov.