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A discontinuous Sobolev function exists


Authors: Przemysław Górka and Artur Słabuszewski
Journal: Proc. Amer. Math. Soc. 147 (2019), 637-639
MSC (2010): Primary 46E35; Secondary 30L99
DOI: https://doi.org/10.1090/proc/14164
Published electronically: October 31, 2018
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Abstract: We prove that there always exist discontinuous functions in the Hajłasz-Sobolev space $ M^{1,s}$ on an $ s$-Ahlfors regular metric space when $ s>1$. In this way an affirmative answer to the conjecture formulated by X. Zhou (2017) is given.


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Przemysław Górka
Affiliation: Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland
Email: pgorka@mini.pw.edu.pl

Artur Słabuszewski
Affiliation: Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland
Email: slabuszewskia@student.mini.pw.edu.pl

DOI: https://doi.org/10.1090/proc/14164
Keywords: Sobolev spaces, metric measure spaces, Ahlfors measure
Received by editor(s): December 12, 2017
Published electronically: October 31, 2018
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2018 American Mathematical Society