The relative $p$-affine capacity

Abstract:

In this paper, the relative $p$-affine capacities are introduced, developed, and subsequently applied to the trace theory of affine Sobolev spaces. In particular, we geometrically characterize such a nonnegative Radon measure $\mu$ given on an open set $\mathcal {O}\subseteq \mathbb R^n$ that naturally induces an embedding of the $p$-affine Sobolev class ${W}^{1,p}_{0,d}(\mathcal {O})$ into the Lebesgue space $L^q(\mathcal {O},\mu )$ (under $1\le p\le q<\infty$) and the exponentially-integrable Lebesgue space $\exp \big ((n\omega _n^\frac 1n|f|)^{n/(n-1)}\big )\in L^1(\mathcal {O},\mu )$ (under $p=n$) as well as the Lebesgue space $L^\infty (\mathcal {O},\mu )$ (under $n<p<\infty$) with $\mu (\mathcal {O})<\infty$. The results discovered here are new and nontrivial.