The relative $p$-affine capacity
HTML articles powered by AMS MathViewer
- by J. Xiao and N. Zhang PDF
- Proc. Amer. Math. Soc. 144 (2016), 3537-3554 Request permission
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.References
- 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, DOI 10.1007/978-3-662-03282-4
- 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
- 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, DOI 10.1007/s00526-009-0235-4
- Christoph Haberl and Franz E. Schuster, Asymmetric affine $L_p$ Sobolev inequalities, J. Funct. Anal. 257 (2009), no. 3, 641–658. MR 2530600, DOI 10.1016/j.jfa.2009.04.009
- Christoph Haberl, Franz E. Schuster, and Jie Xiao, An asymmetric affine Pólya-Szegö principle, Math. Ann. 352 (2012), no. 3, 517–542. MR 2885586, DOI 10.1007/s00208-011-0640-9
- 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
- Alexander Koldobsky, Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005. MR 2132704, DOI 10.1090/surv/116
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17–38. MR 1987375
- 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, DOI 10.1155/IMRN/2006/62987
- 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, DOI 10.1007/978-3-642-15564-2
- Tuo Wang, The affine Sobolev-Zhang inequality on $BV(\Bbb R^n)$, Adv. Math. 230 (2012), no. 4-6, 2457–2473. MR 2927377, DOI 10.1016/j.aim.2012.04.022
- Jie Xiao, The sharp Sobolev and isoperimetric inequalities split twice, Adv. Math. 211 (2007), no. 2, 417–435. MR 2323533, DOI 10.1016/j.aim.2006.08.006
- 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, DOI 10.1007/978-1-4419-1341-8_{1}7
- Jie Xiao, Corrigendum to “The sharp Sobolev and isoperimetric inequalities split twice” [Adv. Math. 211 (2) (2007) 417–435][MR2323533], Adv. Math. 268 (2015), 906–914. MR 3276610, DOI 10.1016/j.aim.2014.04.011
- J. Xiao, The $p$-affine capacity. J. Geometric Anal. DOI: 10.1007/s12220-015-9579-5.
- Jie Xiao and Ning Zhang, Isocapacity estimates for Hessian operators, J. Funct. Anal. 267 (2014), no. 2, 579–604. MR 3210040, DOI 10.1016/j.jfa.2014.04.019
- Zhichun Zhai, Note on affine Gagliardo-Nirenberg inequalities, Potential Anal. 34 (2011), no. 1, 1–12. MR 2740571, DOI 10.1007/s11118-010-9176-y
- Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183–202. MR 1776095
Additional Information
- J. Xiao
- Affiliation: Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland and Labrador A1C 5S7, Canada
- MR Author ID: 247959
- Email: jxiao@mun.ca
- N. Zhang
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 1049706
- Email: nzhang2@ualberta.ca
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3537-3554
- MSC (2010): Primary 53A15, 52A39
- DOI: https://doi.org/10.1090/proc/12980
- MathSciNet review: 3503721