Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions

TL;DR

This paper demonstrates that Hajlasz-Besov and Hajlasz-Triebel-Lizorkin functions with fractional smoothness can be approximated by discrete median convolutions, establishing quasicontinuity and limit properties of medians.

Contribution

It introduces a new approximation method for these functions using median convolutions and proves quasicontinuity of median limits, advancing understanding of their regularity properties.

Findings

Median limits exist quasieverywhere for these functions.

Approximation by median convolutions is possible for Hajlasz-Besov and Hajlasz-Triebel-Lizorkin functions.

Limit of medians defines a quasicontinuous representative.

Abstract

We show that, for $0<s<1$, $0<p<\infty$, $0<q<\infty$, Haj\l asz-Besov and Haj\l asz-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 Haj\l asz 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.