Approximation and quasicontinuity of Besov and Triebel–Lizorkin functions

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, \[ \lim _{r\to 0}m_u^\gamma (B(x,r))=u^*(x), \] 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.