Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access November 8, 2022

Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces

Tomasz Adamowicz , Antoni Kijowski EMAIL logo and Elefterios Soultanis

Abstract

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

References

[1] T. Adamowicz, M. Gaczkowski, P. Górka, Harmonic functions on metric measure spaces, Rev. Mat. Complut. 32(1) (2019), 141–186.10.1007/s13163-018-0272-7Search in Google Scholar

[2] T. Adamowicz, A. Kijowski, E. Soultanis, Asymptotically mean value harmonic functions in subriemannian and RCD settings, submitted.Search in Google Scholar

[3] T. Adamowicz, B. Warhurst, Mean value property and harmonicity on Carnot–Carathéodory groups, Potential Anal. 52 (3) (2020), 497–525. doi.org/10.1007/s11118-018-9740-4.Search in Google Scholar

[4] J. M. Aldaz, Boundedness of averaging operators on geometrically doubling metric spaces Ann. Acad. Sci. Fenn. Math. 44(1) (2019), 497–503.10.5186/aasfm.2019.4430Search in Google Scholar

[5] A. Arroyo, J. Llorente, On the asymptotic mean value property for planar p-harmonic functions, Proc. Amer. Math. Soc. 144(9) (2016), 3859–3868.10.1090/proc/13026Search in Google Scholar

[6] B. Bojarski, P. Hajłasz, P. Strzelecki, Improved Ck, approximation of higher order Sobolev functions in norm and capacity, Indiana Univ. Math. J. 51(3) (2002), 507–540.10.1512/iumj.2002.51.2162Search in Google Scholar

[7] B. Bojarski, L. Ihnatsyeva, J. Kinnunen, How to recognize polynomials in higher order Sobolev spaces Math. Scand. 112(2) (2013), 161–181.10.7146/math.scand.a-15239Search in Google Scholar

[8] A. Bose, Functions satisfying a weighted average property, Trans. Amer. Math. Soc. 118 (1965), 472–487.10.1090/S0002-9947-1965-0177128-0Search in Google Scholar

[9] D. Burago, S. Ivanov, Y. Kurylev, Spectral stability of metric-measure Laplacians Israel J. Math. 232(1) (2019), 125–158.10.1007/s11856-019-1865-7Search in Google Scholar

[10] S. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24(2) (1999), 519–528.Search in Google Scholar

[11] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9(3) (1999), 428–517.10.1007/s000390050094Search in Google Scholar

[12] T. H. Colding, W. P. Minicozzi II, Harmonic Functions on Manifolds, Annals of Math.(2) 146(3), 725–747 (1997).10.2307/2952459Search in Google Scholar

[13] T. H. Colding, W. P. Minicozzi II, Harmonic functions with polynomial growth, J. Differential Geom. 46(1) (1997), 1–77.10.4310/jdg/1214459897Search in Google Scholar

[14] A. Córdoba, J. Ocáriz, A note on generalized laplacians and minimal surfaces, Bull. Lond. Math. Soc. 52 (2020), no. 1, 153–157, doi: 10.1112/blms.12314.10.1112/blms.12314Search in Google Scholar

[15] S. Eriksson-Bique, J. Gill, P. Lahti, N. Shanmugalingam, Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces, Trans. Amer. Math. Soc. 374 (2021), no. 11, 8201–8247Search in Google Scholar

[16] F. Ferrari, Q. Liu, J. Manfredi, On the characterization of p-harmonic functions on the Heisenberg group by mean value properties, Discrete Contin. Dyn. Syst. 34 (7) (2014), 2779–2793.10.3934/dcds.2014.34.2779Search in Google Scholar

[17] F. Ferrari, A. Pinamonti, Characterization by asymptotic mean formulas of q-harmonic functions in Carnot groups, Potential Anal. 42(1) (2015), 203–227.10.1007/s11118-014-9430-9Search in Google Scholar

[18] L. Evans, R. Gariepy, Measure theory and fine properties of functions, Studies in advanced mathematics, CRC Press, Boca Raton, FL, 1992.Search in Google Scholar

[19] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp.10.1007/978-3-642-61798-0Search in Google Scholar

[20] G. Grubb, Distributions and operators, Graduate Texts in Mathematics, 252. Springer, New York, 2009. xii+461 pp.Search in Google Scholar

[21] P. Hajłasz, Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173–218, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.10.1090/conm/338/06074Search in Google Scholar

[22] J. Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.) 44(2) (2007), 163–232.10.1090/S0273-0979-07-01140-8Search in Google Scholar

[23] J. Heinonen, P. Koskela, N. Shanmugalingam, J. Tyson, Sobolev spaces on metric measure spaces. An approach based on upper gradients, New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015.10.1017/CBO9781316135914Search in Google Scholar

[24] B. Hua, M. Kell, C. Xia, Harmonic functions on metric measure spaces, arXiv:1308.3607.Search in Google Scholar

[25] B. Hua, Harmonic functions of polynomial growth on singular spaces with nonnegative Ricci curvature, Proc. Amer. Math. Soc. 139(6) (2011), 2191–2205.10.1090/S0002-9939-2010-10635-4Search in Google Scholar

[26] S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245(2) (2003), 255–292.10.1007/s00209-003-0542-ySearch in Google Scholar

[27] S. Keith, A differentiable structure for metric measure spaces, Adv. Math. 183(2) (2004), 271–315.10.1016/S0001-8708(03)00089-6Search in Google Scholar

[28] A. Kijowski, Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure, Electron. J. Differential Equations 2020, Paper No. 8, 26 pp.Search in Google Scholar

[29] B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth, J. Amer. Math. Soc. 23(3) (2010), 815–829.10.1090/S0894-0347-09-00658-4Search in Google Scholar

[30] B. Kleiner, J. Mackay, Differentiable structures on metric measure spaces: a primer, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16(1) (2016), 41–64.10.2422/2036-2145.201403_004Search in Google Scholar

[31] P. Koskela, D. Yang, Y. Zhou, A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions, J. Funct. Anal. 258(8) (2010), 2637–2661.10.1016/j.jfa.2009.11.004Search in Google Scholar

[32] P. Koskela, D. Yang, Y. Zhou, Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings, Adv. Math. 226(4) (2011), 3579–3621.10.1016/j.aim.2010.10.020Search in Google Scholar

[33] P. Li, Harmonic sections of polynomial growth, Math. Res. Lett. 4(1) (1997), 35–44.10.4310/MRL.1997.v4.n1.a4Search in Google Scholar

[34] P. Li, Harmonic functions and applications to complete manifolds, XIV Escola de Geometria Diferencial. [XIV School of Differential Geometry] Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2006.Search in Google Scholar

[35] J. Manfredi, M. Parviainen, J. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(2) (2012), 215–241.10.2422/2036-2145.201005_003Search in Google Scholar

[36] J. Manfredi, M. Parviainen, J. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal. 42(5) (2010), 2058–2081.10.1137/100782073Search in Google Scholar

[37] A. Minne, D. Tewodrose, Asymptotic Mean Value Laplacian in Metric Measure Spaces, J. Math. Anal. Appl. 491(2) (2020), 124330.10.1016/j.jmaa.2020.124330Search in Google Scholar

[38] A. Minne, D. Tewodrose, Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces, arXiv:2202.09295.Search in Google Scholar

[39] D. Yang, New characterizations of Hajłasz-Sobolev spaces on metric spaces, Sci. China Ser. A 46(5) (2003), 675–689.10.1360/02ys0343Search in Google Scholar

Received: 2022-06-14
Accepted: 2022-08-16
Published Online: 2022-11-08

© 2022 Tomasz Adamowicz et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 2.2.2023 from https://www.degruyter.com/document/doi/10.1515/agms-2022-0143/html
Scroll Up Arrow