On limiting trace inequalities for vectorial differential operators

Abstract

We establish that trace inequalities $$\|D^{k-1}u\|_{L^{\frac{n-s}{n-1}}(\mathbb{R}^{n},d\mu)} \leq c \|\mu\|_{L^{1,n-s}(\mathbb{R}^{n})}^{\frac{n-1}{n-s}}\|\mathbb{A}[D]u\|_{L^{1}(\mathbb{R}^{n},d\mathscr{L}^{n})}$$ hold for vector fields $u\in C^{\infty}(\mathbb{R}^{n};\mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $\mathbb{A}[D]$ on $\mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $\mathbb{A}[D]$ are necessary. Here, $\|\mu\|_{L^{1,\lambda}}$ denotes the $(1,\lambda)$-Morrey norm of the measure $\mu$, so that such traces can be taken, for example, with respect to the Hausdorff measure $\mathscr{H}^{n-s}$ restricted to fractals of codimension $0<s<1$. The above class of inequalities give a systematic generalisation of Adams' trace inequalities to the limit case $p=1$ and can be used to prove trace embeddings for functions of bounded $\mathbb{A}$-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We moreover establish a multiplicative version of the above inequality, which implies ($\mathbb{A}$-)strict continuity of the associated trace operators on $\text{BV}^{\mathbb{A}}$.