Abstract
We investigate weighted Sobolev spaces on metric measure spaces
Funding statement: The authors acknowledge the support of the grant ERC ADG GeMeThNES. The second author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Acknowledgements
We would like to thank M. D. Surnachev for kindly explaining the change of weight in [27] (which we adapted in Proposition 4.4) and for informing us of his generalization of Zhikov’s result to more general exponents [26].
References
[1] L. Ambrosio, M. Colombo and S. Di Marino, Sobolev spaces in metric measure spaces: Reflexivity and lower semicontinuity of slope, Adv. Stud. Pure Math. 67 (2015), 1–58. Search in Google Scholar
[2] L. Ambrosio and S. Di Marino, Equivalent definitions of BV space and of total variation on metric measure spaces, J. Funct. Anal. 266 (2014), 4150–4188. 10.1016/j.jfa.2014.02.002Search in Google Scholar
[3] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd ed., Lectures in Math. ETH Zürich, Birkhäuser, Basel 2008. Search in Google Scholar
[4] L. Ambrosio, N. Gigli and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam. 29 (2013), 269–286. 10.4171/RMI/746Search in Google Scholar
[5] L. Ambrosio, N. Gigli and G. Savaré, Calculus and heat flow on metric measure spaces and applications to spaces with Ricci curvature bounds from below, Invent. Math. 195 (2014), 289–391. 10.1007/s00222-013-0456-1Search in Google Scholar
[6]
L. Ambrosio, A. Pinamonti and G. Speight,
Tensorization of Cheeger energies, the space
[7] R. Bhatia, Matrix analysis, Grad. Texts in Math. 169, Springer, New York 2007. Search in Google Scholar
[8] A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts Math. 17, European Mathematical Society, Zürich 2011. 10.4171/099Search in Google Scholar
[9] J. Björn, S. Buckley and S. Keith, Admissible measures in one dimension, Proc. Amer. Math. Soc. 134 (2006), 703–705. 10.1090/S0002-9939-05-07925-6Search in Google Scholar
[10] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428–517. 10.1007/s000390050094Search in Google Scholar
[11] V. Chiadó Piat and F. Serra Cassano, Relaxation of degenerate variational integrals, Nonlinear Anal. 22 (1994), 409–424. 10.1016/0362-546X(94)90165-1Search in Google Scholar
[12] V. Chiadó Piat and F. Serra Cassano, Some remarks about the density of smooth functions in weighted Sobolev spaces, J. Convex Anal. 1 (1994), 135–142. Search in Google Scholar
[13] S. Di Marino and G. Speight, The p-weak gradient depends on p, Proc. Amer. Math. Soc. 143 (2015), 5239–5252. 10.1090/S0002-9939-2015-12641-XSearch in Google Scholar
[14] B. Franchi, P. Hajlasz and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1903–1924. 10.5802/aif.1742Search in Google Scholar
[15] P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 688 (2000), 1–101. 10.1090/memo/0688Search in Google Scholar
[16] J. Heinonen, Lectures on analysis on metric spaces, Springer, New York 2001. 10.1007/978-1-4613-0131-8Search in Google Scholar
[17] J. Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc. 44 (2007), 163–232. 10.1090/S0273-0979-07-01140-8Search in Google Scholar
[18] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Clarendon Press, Oxford 1993. Search in Google Scholar
[19] J. Heinonen, P. Koskela, N. Shanmugalingam and J. Tyson, Sobolev spaces on metric measure spaces: An approach based on upper gradients, New Math. Monogr. 27, Cambridge University Press, Cambridge 2015. 10.1017/CBO9781316135914Search in Google Scholar
[20] S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245 (2003), 255–292. 10.1007/s00209-003-0542-ySearch in Google Scholar
[21] S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), 575–599. 10.4007/annals.2008.167.575Search in Google Scholar
[22] B. Kleiner and J. Mackay, Differentiable structures on metric measure spaces: A primer, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 1, 41–64. Search in Google Scholar
[23] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. 10.1090/S0002-9947-1972-0293384-6Search in Google Scholar
[24] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279. 10.4171/RMI/275Search in Google Scholar
[25] J. O. Strömberg and A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Math. 1381, Springer, Berlin 1989. 10.1007/BFb0091154Search in Google Scholar
[26] M. D. Surnachev, Density of smooth functions in weighted Sobolev spaces with variable exponents, Dokl. Math. 89 (2014), 146–150. 10.1134/S1064562414020045Search in Google Scholar
[27] V. V. Zhikov, Density of smooth functions in weighted Sobolev spaces, Dokl. Math. 88 (2013), 669–673. 10.1134/S1064562413060173Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
