Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 18, 2016

Weighted Sobolev spaces on metric measure spaces

  • Luigi Ambrosio EMAIL logo , Andrea Pinamonti and Gareth Speight


We investigate weighted Sobolev spaces on metric measure spaces (X,d,𝔪). Denoting by ρ the weight function, we compare the space W1,p(X,d,ρ𝔪) (which always coincides with the closure H1,p(X,d,ρ𝔪) of Lipschitz functions) with the weighted Sobolev spaces Wρ1,p(X,d,𝔪) and Hρ1,p(X,d,𝔪) defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W1,p(X,d,ρ𝔪)=Hρ1,p(X,d,𝔪). We also adapt the results in [23] and in the recent paper [27] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality Wρ1,p(X,d,𝔪)=Hρ1,p(X,d,𝔪).

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).


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].


[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 W1,1 and the area formula, Adv. Math. 281 (2015), 1145–1177. 10.1016/j.aim.2015.06.004Search in Google Scholar

[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

Received: 2015-03-24
Revised: 2016-01-27
Published Online: 2016-05-18
Published in Print: 2019-01-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 3.2.2023 from
Scroll Up Arrow