Summary: We establish that trace inequalities for vector fields \(u\in\mathrm{C}^{\infty}_c(\mathbb{R}^n,\mathbb{R}^N)\)\[\|D^{k-1}u\|_{L^{(n-s)/(n-1)}(\mathrm{d}\mu)} \leqslant c\big\|\mu\big\|^{(n-1)/(n-s)}_{L^{1,n-s}} \big\|\mathbb{A}[D]u\big\|_{L^1(\mathrm{d}\mathcal{L}^n)} \tag{\(*\)}\]hold 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 we give partial results for \(s=1\), where stronger conditions on \(\mathbb{A}[D]\) are necessary. Here, \(\|\mu\|_{L^{1,\lambda}}\) denotes the Morrey norm of \(\mu\) so that such traces can be taken, for example, with respect to \(\mathcal{H}^{n-s} \)-measure restricted to fractals of codimension \(s<1\). The class of inequalities \((*)\) gives a systematic generalisation of Adams’s 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 also prove a multiplicative version of \((*)\), which implies strict continuity of the associated trace operators on \(\mathrm{BV}^{\mathbb{A}} \).

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