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Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions


Authors: Toni Heikkinen, Pekka Koskela and Heli Tuominen
Journal: Trans. Amer. Math. Soc. 369 (2017), 3547-3573
MSC (2010): Primary 46E35, 43A85
DOI: https://doi.org/10.1090/tran/6886
Published electronically: December 22, 2016
MathSciNet review: 3605979
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Abstract: We show that, for $ 0<s<1$, $ 0<p<\infty $, $ 0<q<\infty $, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions can be approximated in the norm by discrete median convolutions. This allows us to show that, for these functions, the limit of medians,

$\displaystyle \lim _{r\to 0}m_u^\gamma (B(x,r))=u^*(x), $

exists quasieverywhere and defines a quasicontinuous representative of $ u$. The above limit exists quasieverywhere also for Hajłasz functions $ u\in M^{s,p}$, $ 0<s\le 1$, $ 0<p<\infty $, but approximation of $ u$ in $ M^{s,p}$ by discrete (median) convolutions is not in general possible.

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Additional Information

Toni Heikkinen
Affiliation: Department of Mathematics and Systems Analysis, P.O. Box 11100, FI-00076 Aalto University, Finland
Email: toni.heikkinen@aalto.fi

Pekka Koskela
Affiliation: Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland
Email: pekka.j.koskela@jyu.fi

Heli Tuominen
Affiliation: Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland
Email: heli.m.tuominen@jyu.fi

DOI: https://doi.org/10.1090/tran/6886
Keywords: Besov space, Triebel--Lizorkin space, fractional Sobolev space, metric measure space, median, quasicontinuity
Received by editor(s): May 21, 2015
Published electronically: December 22, 2016
Article copyright: © Copyright 2016 American Mathematical Society