On the sharp lower bound for duality of modulus
- 24 March 2022
- journal article
- research article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 150 (7), 2955-2968
- https://doi.org/10.1090/proc/15951
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.
Keywords
Funding Information
- Academy of Finland (330048)
This publication has 21 references indexed in Scilit:
- Measurability of equivalence classes and MEC$_p$-property in metric spacesRevista Matemática Iberoamericana, 2007
- Subsets of rectifiable curves in Hilbert space-the analyst’s TSPJournal d'Analyse Mathématique, 2007
- Newtonian spaces: An extension of Sobolev spaces to metric measure spacesRevista Matemática Iberoamericana, 2000
- Differentiability of Lipschitz Functions on Metric Measure SpacesGeometric and Functional Analysis, 1999
- Quasiconformal maps in metric spaces with controlled geometryActa Mathematica, 1998
- Extension of functions preserving the modulus of continuityMathematical Notes, 1997
- Rectifiable Metric Spaces: Local Structure and Regularity of the Hausdorff MeasureProceedings of the American Mathematical Society, 1994
- Extremal Length and Conformal CapacityTransactions of the American Mathematical Society, 1967
- Conformal invariants and function-theoretic null-setsActa Mathematica, 1950
- Continua of Finite Linear Measure IAmerican Journal of Mathematics, 1943