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Differential expressions with mixed homogeneity and spaces of smooth functions they generate in arbitrary dimension. (English) Zbl 1348.46037
Summary: Let $$\{T_1, \dots, T_J \}$$ 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 $$T_j f$$, $$j = 1, \dots, J$$, are continuous. Extending the previous work of the first two authors, we analyze 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) $$\{\sigma_1, \dots, \sigma_J \}$$ from the initial operators $$\{T_1, \dots, T_J \}$$. Let $$K$$ be the dimension of the linear span of $$\{\sigma_1, \dots, \sigma_J \}$$. If $$K \geqslant 2$$, 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 anisotropic embedding theorem of Sobolev type for vector fields.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46B25 Classical Banach spaces in the general theory
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