On the sharp lower bound for duality of modulus

Eriksson-Bique, Sylvester; Poggi-Corradini, Pietro

doi:10.1090/proc/15951

On the sharp lower bound for duality of modulus
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by Sylvester Eriksson-Bique and Pietro Poggi-Corradini PDF

Proc. Amer. Math. Soc. 150 (2022), 2955-2968 Request permission

Abstract:

We establish a sharp reciprocity inequality for modulus in compact metric spaces $X$ with finite Hausdorff measure. In particular, when $X$ is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney [Ann. Acad. Sci. Fenn. Math. 44 (2019), pp. 681-692]. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua $E$ and $F$ in $X$ and the modulus of the family of surfaces of finite Hausdorff measure that separate $E$ and $F$. The paper also develops approximation techniques, which may be of independent interest.

References

Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129. MR 36841, DOI 10.1007/BF02392634
Hiroaki Aikawa and Makoto Ohtsuka, Extremal length of vector measures, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 1, 61–88. MR 1677985
Nathan Albin, Jason Clemens, Nethali Fernando, and Pietro Poggi-Corradini, Blocking duality for $p$-modulus on networks and applications, Ann. Mat. Pura Appl. (4) 198 (2019), no. 3, 973–999. MR 3954401, DOI 10.1007/s10231-018-0806-0
Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127–183. MR 1930885, DOI 10.1007/s00222-002-0233-z
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. MR 1708448, DOI 10.1007/s000390050094
Claude Dellacherie, Ensembles analytiques, capacités, mesures de Hausdorff, Lecture Notes in Mathematics, Vol. 295, Springer-Verlag, Berlin-New York, 1972 (French). MR 0492152, DOI 10.1007/BFb0060706
Samuel Eilenberg and O. G. Harrold Jr., Continua of finite linear measure. I, Amer. J. Math. 65 (1943), 137–146. MR 7643, DOI 10.2307/2371777
Sylvester Eriksson-Bique, Density of Lipschitz functions in energy, Preprint, arXiv:2012.01892, 2020.
Behnam Esmayli and Piotr Hajłasz, The coarea inequality, Ann. Fenn. Math. 46 (2021), no. 2, 965–991. MR 4307012, DOI 10.5186/aasfm.2021.4654
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson, Sobolev spaces on metric measure spaces, New Mathematical Monographs, vol. 27, Cambridge University Press, Cambridge, 2015. An approach based on upper gradients. MR 3363168, DOI 10.1017/CBO9781316135914
Esa Järvenpää, Maarit Järvenpää, Kevin Rogovin, Sari Rogovin, and Nageswari Shanmugalingam, Measurability of equivalence classes and $\textrm {MEC}_p$-property in metric spaces, Rev. Mat. Iberoam. 23 (2007), no. 3, 811–830. MR 2414493, DOI 10.4171/RMI/514
Rebekah Jones and Panu Lahti, Duality of moduli and quasiconformal mappings in metric spaces, Anal. Geom. Metr. Spaces 8 (2020), no. 1, 166–181. MR 4141371, DOI 10.1515/agms-2020-0112
Stephen Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245 (2003), no. 2, 255–292. MR 2013501, DOI 10.1007/s00209-003-0542-y
Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113–123. MR 1189747, DOI 10.1090/S0002-9939-1994-1189747-7
Atte Lohvansuu and Kai Rajala, Duality of moduli in regular metric spaces, Indiana Univ. Math. J. 70 (2021), no. 3, 1087–1102. MR 4284107, DOI 10.1512/iumj.2021.70.8433
V. A. Mil′man, Extension of functions that preserve the modulus of continuity, Mat. Zametki 61 (1997), no. 2, 236–245 (Russian, with Russian summary); English transl., Math. Notes 61 (1997), no. 1-2, 193–200. MR 1620006, DOI 10.1007/BF02355728
M. H. A. Newman, Elements of the topology of plane sets of points, Cambridge University Press, New York, 1961. Second edition, reprinted. MR 0132534
Kai Rajala, Uniformization of two-dimensional metric surfaces, Invent. Math. 207 (2017), no. 3, 1301–1375. MR 3608292, DOI 10.1007/s00222-016-0686-0
Kai Rajala and Matthew Romney, Reciprocal lower bound on modulus of curve families in metric surfaces, Ann. Acad. Sci. Fenn. Math. 44 (2019), no. 2, 681–692. MR 3973535, DOI 10.5186/aasfm.2019.4442
Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
Raanan Schul, Subsets of rectifiable curves in Hilbert space—the analyst’s TSP, J. Anal. Math. 103 (2007), 331–375. MR 2373273, DOI 10.1007/s11854-008-0011-y
Nageswari Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243–279. MR 1809341, DOI 10.4171/RMI/275
William P. Ziemer, Extremal length and conformal capacity, Trans. Amer. Math. Soc. 126 (1967), 460–473. MR 210891, DOI 10.1090/S0002-9947-1967-0210891-0