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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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