Topological Hausdorff dimension and Poincaré inequality
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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.
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How to Cite
DiMarco, C. (2022). Topological Hausdorff dimension and Poincaré inequality. Extracta Mathematicae, 37(2), 211-221. https://doi.org/10.17398/2605-5686.37.2.211
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Topology
References
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[2] A. Björn, J. Björn, “ Nonlinear Potential Theory on Metric Spaces ”, EMS Tracts in Mathematics 17, European Mathematical Society (EMS), Zürich, 2011.
[3] K. Falconer, “ Fractal Geometry ”, Second edition, Mathematical foundations and applications, John Wiley & Sons, Inc., Hoboken, NJ, 2003.
[4] J. Heinonen, “ Lectures on Analysis on Metric Spaces ”, Universitext, Springer-Verlag, New York, 2001.
[5] 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.
[6] A. Lohvansuu, K. Rajala, Duality of moduli in regular metric spaces, Indiana Univ. Math. J. 70 (3) (2021), 1087 – 1102.
[7] H. Lotfi, The µ-topological Hausdorff dimension, Extracta Math. 34 (2) (2019), 237 – 254.
[8] 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.