Approximation and Quasicontinuity of Besov and Triebel–Lizorkin Functions
Heikkinen, T., Koskela, P., & Tuominen, H. (2017). Approximation and Quasicontinuity of Besov and Triebel–Lizorkin Functions. Transactions of the American Mathematical Society, 369 (5), 3547-3573. doi:10.1090/tran/6886
Julkaistu sarjassaTransactions of the American Mathematical Society
© 2016 American Mathematical Society. This is a final draft version of an article whose final and definitive form has been published by AMS. Published in this repository with the kind permission of the publisher.
We show that, for 0 < s < 1, 0 < p, q < ∞, Haj lasz–Besov and Haj lasz–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, lim r→0 mγ u (B(x, r)) = u ∗ (x), exists quasieverywhere and defines a quasicontinuous representative of u. The above limit exists quasieverywhere also for Haj lasz 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.
JulkaisijaAmerican Mathematical Society