Approximation and quasicontinuity of Besov and Triebel–Lizorkin functions

Toni Heikkinen, Pekka Koskela, Heli Tuominen

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)

Abstract

We show that, for 0 < s < 1, 0 < p < ∞, 0 < q < ∞, 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, [Formula Presented] exists quasieverywhere and defines a quasicontinuous representative of u. The above limit exists quasieverywhere also for Hajłasz functions u ∈ Ms,p, 0 < s ≤ 1, 0 < p <∞, but approximation of u in Ms,p by discrete (median) convolutions is not in general possible.

Original languageEnglish
Pages (from-to)3547-3573
Number of pages27
JournalTransactions of the American Mathematical Society
Volume369
Issue number5
DOIs
Publication statusPublished - 2017
MoE publication typeA1 Journal article-refereed

Keywords

  • Besov space
  • Fractional Sobolev space
  • Median
  • Metric measure space
  • Quasicontinuity
  • Triebel–Lizorkin space

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