Approximation and Quasicontinuity of Besov and Triebel–Lizorkin Functions

Abstract

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.