Year: 2021
Type: article
Abstract: We establish that trace inequalities $$\|D^{k-1}u\|_{L^{\frac{n-s}{n-1}}(\mathbb{R}^{n},dμ)} \leq c \|μ\|_{L^{1,n-s}(\mathbb{R}^{n})}^{\frac{n-1}{n-s}}\|\mathbb{A}[D]u\|_{L^{1}(\mathbb{R}^{n},d\mathscr{L}^{n})}$$ 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, $\|μ\|_{L^{1,λ}}$ denotes the $(1,λ)$-Morrey norm of the measure $μ$, 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
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2922592508