Topological Hausdorff dimension and Poincaré inequality

Authors

  • C.A. DiMarco 1000 E. Henrietta Rd., Mathematics Department, Monroe Community College Rochester, NY 14623, USA

DOI:

https://doi.org/10.17398/2605-5686.37.2.211

Keywords:

Poincaré inequality, metric space, Cantor sets, topological dimension, Hausdorff dimension, bi-Lipschitz map, Ahlfors regular

Abstract

A relationship between Poincaré inequalities and the topological Hausdorff dimension is exposed—a lower bound on the dimension of Ahlfors regular spaces satisfying a weak (1, p)-Poincaré inequality is given.

Downloads

Download data is not yet available.

References

R. Balka, Z. Buczolich, M. Elekes, A new fractal dimension: the topological Hausdorff dimension, Adv. Math. 274 (2015), 881 – 927.

A. Björn, J. Björn, “ Nonlinear Potential Theory on Metric Spaces ”, EMS Tracts in Mathematics 17, European Mathematical Society (EMS), Zürich, 2011.

K. Falconer, “ Fractal Geometry ”, Second edition, Mathematical foundations and applications, John Wiley & Sons, Inc., Hoboken, NJ, 2003.

J. Heinonen, “ Lectures on Analysis on Metric Spaces ”, Universitext, Springer-Verlag, New York, 2001.

J. Heinonen, P. Koskela, N. Shanmugalingam, J.T. Tyson, “ Sobolev Spaces on Metric Measure Spaces. An Approach based on Upper Gradients ”, New Mathematical Monographs 27, Cambridge University Press, Cambridge, 2015.

A. Lohvansuu, K. Rajala, Duality of moduli in regular metric spaces, Indiana Univ. Math. J. 70 (3) (2021), 1087 – 1102.

H. Lotfi, The µ-topological Hausdorff dimension, Extracta Math. 34 (2) (2019), 237 – 254.

J.M. Mackay, J.T. Tyson, K. Wildrick, Modulus and Poincaré inequalities on non-self-similar Sierpiński carpets, Geom. Funct. Anal., 23 (3) (2013), 985 – 1034.

Downloads

Published

2022-12-01

Issue

Section

Topology

How to Cite

Topological Hausdorff dimension and Poincaré inequality. (2022). Extracta Mathematicae, 37(2), 211-221. https://doi.org/10.17398/2605-5686.37.2.211