Embeddings for $\mathbb{A}$-weakly differentiable functions on domains

Abstract

We prove that the critical embedding ${\mathrm{W}}^{\mathbb{A},1}(B)\hookrightarrow {\mathrm{W}}^{k-1,\frac{n}{n-1}}$ holds if and only if the $k$-homogeneous, linear differential operator $\mathbb{A}$ on ${\mathbb{R}}^n$ from ${\mathbb{R}}^N$ to ${\mathbb{R}}^m$ has finite dimensional null-space. Here $B$ is a ball in ${\mathbb{R}}^n$ and ${\mathrm{W}}^{\mathbb{A},1}(B)$ denotes the space of maps $u\in {\mathrm{L}}^1(B,{\mathbb{R}}^N)$ such that the vector valued distribution $\mathbb{A}u$ is an integrable map. The result was previously known only for several examples of $\mathbb{A}$. Our result contrasts the homogeneous embedding in full-space. Namely, Van Schaftingen proved that $\dot{\mathrm{W}}{}^{\mathbb{A},1}({\mathbb{R}}^n)\hookrightarrow \dot{\mathrm{W}}{}^{k-1,\frac{n}{n-1}}$ if and only if $\mathbb{A}$ is elliptic and cancelling. We show that this condition is (strictly) implied by $\mathbb{A}$ having finite dimensional null-space.

23 pages, 1 table

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