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The relative $ p$-affine capacity

Authors: J. Xiao and N. Zhang
Journal: Proc. Amer. Math. Soc. 144 (2016), 3537-3554
MSC (2010): Primary 53A15, 52A39
Published electronically: March 1, 2016
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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\vert f\vert)^{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.

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J. Xiao
Affiliation: Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland and Labrador A1C 5S7, Canada

N. Zhang
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Keywords: Relative $p$-affine capacity, Radon measure, space embedding, $p$-affine Sobolev class
Received by editor(s): May 5, 2015
Received by editor(s) in revised form: September 29, 2015
Published electronically: March 1, 2016
Additional Notes: This project was supported by NSERC of Canada as well as by URP of Memorial University, Canada.
Communicated by: Guofang Wei
Article copyright: © Copyright 2016 American Mathematical Society