Approximation by H\"older functions in Besov and Triebel-Lizorkin spaces

TL;DR

This paper demonstrates that functions in Besov and Triebel-Lizorkin spaces can be approximated by H"older continuous functions in metric measure spaces, using new median inequalities and pointwise inequalities.

Contribution

It introduces new approximation results for Besov and Triebel-Lizorkin functions via H"older functions in metric measure spaces, including novel median inequalities.

Findings

Approximation of Besov and Triebel-Lizorkin functions by H"older functions in metric spaces.

New median inequalities, including a Poincaré type inequality, established.

Proofs valid for Haj{ }asz-Besov and Haj{ }asz-Triebel-Lizorkin functions.

Abstract

In this paper, we show that Besov and Triebel-Lizorkin functions can be approximated by a H\"older continuous function both in the Lusin sense and in norm. The results are proven in metric measure spaces for Haj{\l}asz-Besov and Haj{\l}asz-Triebel-Lizorkin functions defined by a pointwise inequality. We also prove new inequalities for medians, including a Poincar\'e type inequality, which we use in the proof of the main result.