On limiting trace inequalities for vectorial differential operators

Abstract

We establish that trace inequalities $$\Vert D^{k-1}u{\Vert }_{L^{\frac{n-s}{n-1}}({\mathbb{R}}^n,d\mu )}\le c\Vert \mu {\Vert }_{L^{1,n-s}({\mathbb{R}}^n)}^{\frac{n-1}{n-s}}\Vert \mathbb{A}[D]u{\Vert }_{L^1({\mathbb{R}}^n,d{\mathcal{L}}^n)}$$ 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, $\Vert \mu {\Vert }_{L^{1,\lambda }}$ 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 ${\mathcal{H}}^{n-s}$ 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 ${\text{BV}}^{\mathbb{A}}$.

30 pages, 4 figures

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• On dimension and regularity of bundle measures
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• $\mathbb{C}$-elliptic operators and ${\mathrm{W}}^{1,1}$-regularity for linear growth functionals
• New Directions in Harmonic Analysis on $L^1$
• Regularity for the Dirichlet problem on BD
• Continuity points via Riesz potentials for $\mathbb{C}$-elliptic operators
• Syzygies, constant rank, and beyond
• Intrinsic nature of the Stein-Weiss $H^1$-inequality