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

Let {T1,…,TJ}{T1,…,TJ} be a collection of differential operators with constant coefficients on the torus TnTn. Consider the Banach space X of functions f   on the torus for which all functions TjfTjf, j=1,…,Jj=1,…,J, are continuous. Extending the previous work of the first two authors, we analyze the embeddability of X   into some space C(K)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) {σ1,…,σJ}{σ1,…,σJ} from the initial operators {T1,…,TJ}{T1,…,TJ}. Let K   be the dimension of the linear span of {σ1,…,σJ}{σ1,…,σJ}. If K⩾2K⩾2, then X   is not isomorphic to a complemented subspace of C(K)C(K) for any compact space K.The main ingredient of the proof of this fact is a new anisotropic embedding theorem of Sobolev type for vector fields.