Approximation and quasicontinuity of Besov and Triebel–Lizorkin functions

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 ∈ M^s,p, 0 < s ≤ 1, 0 < p <∞, but approximation of u in M^s,p by discrete (median) convolutions is not in general possible.