Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/proc/12980
Published electronically: March 1, 2016
MathSciNet review: 3503721
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441 (97j:46024)
  • [2] Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195-199. MR 0402831 (53 #6645)
  • [3] Andrea Cianchi, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 3, 419-436. MR 2551138 (2010h:46041), https://doi.org/10.1007/s00526-009-0235-4
  • [4] Christoph Haberl and Franz E. Schuster, Asymmetric affine $ L_p$ Sobolev inequalities, J. Funct. Anal. 257 (2009), no. 3, 641-658. MR 2530600 (2010j:46068), https://doi.org/10.1016/j.jfa.2009.04.009
  • [5] Christoph Haberl, Franz E. Schuster, and Jie Xiao, An asymmetric affine Pólya-Szegö principle, Math. Ann. 352 (2012), no. 3, 517-542. MR 2885586, https://doi.org/10.1007/s00208-011-0640-9
  • [6] Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810 (94e:31003)
  • [7] Alexander Koldobsky, Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005. MR 2132704 (2006a:42007)
  • [8] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $ L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17-38. MR 1987375 (2004d:46039)
  • [9] Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Optimal Sobolev norms and the $ L^p$ Minkowski problem, Int. Math. Res. Not. , posted on (2006), Art. ID 62987, 21. MR 2211138 (2007d:52007), https://doi.org/10.1155/IMRN/2006/62987
  • [10] Vladimir Maz'ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530 (2012a:46056)
  • [11] Tuo Wang, The affine Sobolev-Zhang inequality on $ BV(\mathbb{R}^n)$, Adv. Math. 230 (2012), no. 4-6, 2457-2473. MR 2927377, https://doi.org/10.1016/j.aim.2012.04.022
  • [12] Jie Xiao, The sharp Sobolev and isoperimetric inequalities split twice, Adv. Math. 211 (2007), no. 2, 417-435. MR 2323533 (2008k:46108), https://doi.org/10.1016/j.aim.2006.08.006
  • [13] Jie Xiao, The $ p$-Faber-Krahn inequality noted, Around the research of Vladimir Maz'ya. I, Int. Math. Ser. (N. Y.), vol. 11, Springer, New York, 2010, pp. 373-390. MR 2723828 (2011e:35135), https://doi.org/10.1007/978-1-4419-1341-8_17
  • [14] Jie Xiao, Corrigendum to ``The sharp Sobolev and isoperimetric inequalities split twice'' [Adv. Math. 211 (2) (2007) 417-435][MR 2323533], Adv. Math. 268 (2015), 906-914. MR 3276610, https://doi.org/10.1016/j.aim.2014.04.011
  • [15] J. Xiao, The $ p$-affine capacity. J. Geometric Anal. DOI: 10.1007/s12220-015-9579-5.
  • [16] Jie Xiao and Ning Zhang, Isocapacity estimates for Hessian operators, J. Funct. Anal. 267 (2014), no. 2, 579-604. MR 3210040, https://doi.org/10.1016/j.jfa.2014.04.019
  • [17] Zhichun Zhai, Note on affine Gagliardo-Nirenberg inequalities, Potential Anal. 34 (2011), no. 1, 1-12. MR 2740571 (2011j:46060), https://doi.org/10.1007/s11118-010-9176-y
  • [18] Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202. MR 1776095 (2001m:53136)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53A15, 52A39

Retrieve articles in all journals with MSC (2010): 53A15, 52A39


Additional Information

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

N. Zhang
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: nzhang2@ualberta.ca

DOI: https://doi.org/10.1090/proc/12980
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

American Mathematical Society