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

TL;DR

This paper investigates the structure of function spaces defined by differential operators with mixed homogeneity on the torus, proving non-embeddability into spaces of continuous functions under certain conditions using a new Sobolev-type embedding theorem.

Contribution

It extends previous work by analyzing embeddability of these function spaces with mixed homogeneity, establishing non-isomorphism to complemented subspaces of continuous functions when the span of homogeneous parts has dimension at least two.

Findings

If the span of the homogeneous parts has dimension ≥ 2, the space is not complemented in any C(K) space.

Introduces a new Sobolev-type embedding theorem for these function spaces.

Provides conditions under which the space cannot be embedded into continuous function spaces.

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) ${tau_1,...,tau_l}$ from the initial operators ${T_1,...,T_l}$. Let $N$ be the dimension of the linear span of ${\tau_1,...,\tau_l}$. If $N\geqslant 2$, 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 Sobolev-type embedding theorem.