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

TL;DR

This paper characterizes when certain embeddings of weakly differentiable functions hold, showing they are equivalent to the differential operator having a finite dimensional null-space, extending previous results known only for specific cases.

Contribution

It establishes a new necessary and sufficient condition for embeddings of $ ext{W}^{ ext{A},1}$ spaces, linking finite dimensional null-space of the operator to the embedding property.

Findings

Embedding holds iff $ ext{A}$ has finite dimensional null-space.

Contrasts with full-space homogeneous embedding results.

Finite dimensional null-space implies ellipticity and cancellation.

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.