Wavelet decomposition techniques and Hardy inequalities for function spaces on cellular domains

TL;DR

This paper investigates wavelet basis construction on cellular domains for critical function spaces, extending Triebel's work by analyzing reinforced spaces to include exceptional smoothness parameters.

Contribution

It extends Triebel's wavelet basis construction to critical cases using reinforced function spaces, filling a gap in the theory for Sobolev and Besov spaces.

Findings

Established decomposition theorems for reinforced Triebel-Lizorkin spaces

Extended wavelet basis construction to critical smoothness parameters

Provided new tools for analyzing function spaces on cellular domains

Abstract

A rather tricky question is the construction of wavelet bases on domains for suitable function spaces (Sobolev, Besov, Triebel-Lizorkin type). In his monograph from 2008, Triebel presented an approach how to construct wavelet (Riesz) bases in function spaces of Besov and Triebel-Lizorkin type on cellular domains, in particular on the cube. However, he had to exclude essential exceptional values of the smoothness parameter $s$, for instance the theorems do not cover the Sobolev space W_2^1(Q) on the n-dimensional cube Q for n at least 2. Triebel also gave an idea how to deal with those exceptional values for the Triebel-Lizorkin function space scale on the cube Q: He suggested to introduce modified function spaces for the critical values, the so-called reinforced spaces. In this paper we start examining these reinforced spaces and transfer the crucial decomposition theorems necessary for establishing a wavelet basis from the non-critical values to analogous results for the critical cases now decomposing the reinforced function spaces of Triebel-Lizorkin type.