Abstract
In this paper we formulate some basic properties for the BV capacity and Hausdorff capacity on the Grushin space \(\mathbb {G}^n_{\alpha }\) and develop the sharp BV isocapacity inequalities on \(\mathbb {G}^n_{\alpha }\) under a dimensional condition.
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D. R. Adams, A note on Choquet integral with respect to Hausdorff capacity, In: Function Spaces and Applications, 115-124, Lund 1986, Lecture Notes in Math., 1302, Springer-Verlag, Berlin, 1988.
L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces. Calculus of variations, nonsmooth analysis and related topics, Set-Valued Anal. 10 (2002), 111–128.
B. Franchi and E. Lanconelli, Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Conference on linear partial and pseudodifferential operators, (Torino, 1982), Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, 105–114.
B. Franchi, BV spaces and rectifiability for Carnot–Carathéodory metrics: an introduction, In: NAFSA 7–Nonlinear Analysis, Vol. 7, 72–132, Czech. Acad. Sci., Prague, 2003.
N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces, Commun. Pure Appl. Math. 49 (1996), 1081–1144.
J. Heinonen, Lectures on Analysis on Metric Spaces, Springer-Verlag, New York, 2001.
H. Hakkarainen and J. Kinnunen, The BV capacity in metric spaces, Manuscripta Math. 132 (2010), 51–73.
Y. Liu, BV capacity on generalized Grushin plane, J. Geom. Anal. 27 (2017), 409–441.
Yu Liu and J. Xiao, Functional capacities on the Grushin space \(\mathbb{G}^n_{\alpha }\), Annali di Matematica (2017). https://doi.org/10.1007/s10231-017-0699-3
J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, Lebesgue points and capacities via boxing inequality in metric spaces, Indiana Univ. Math. J. 57 (2008), 401–430.
M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. 82 (2003), 975–1004.
V. Franceschi and R. Monti, Isoperimetric in \(H\)-type groups and Grushin spaces, Rev. Mat. Iberoam. 32 (2016), 1227–1258.
F. Montefalcone, Sets of finite perimeter associated with vector fields and polyhedral approximation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), 4 (2004), 279–295.
R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane, J. Geom. Anal. 14 (2004), 355–368.
R. Monti and F. S. Cassano, Surface measures in Carnot–Carathéodory spaces, Calc. Var. 13 (2001), 339–376.
J. Xiao, The sharp Sobolev and isoperimetric inequalities split twice, Adv. Math. 211 (2007), 417–435.
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Supported by the National Natural Science Foundation of China (Nos. 11671031, 11471018), the Fundamental Research Funds for the Central Universities (No. FRF-BR-17-004B), and Beijing Municipal Science and Technology Project (No. Z17111000220000).
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Li, G., Liu, Y. A note on BV capacities on Grushin spaces. Arch. Math. 111, 203–214 (2018). https://doi.org/10.1007/s00013-018-1174-0
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DOI: https://doi.org/10.1007/s00013-018-1174-0