Wavelet decomposition techniques and Hardy inequalities for 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 , 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.
Keywords
Cite
@article{arxiv.1306.3131,
title = {Wavelet decomposition techniques and Hardy inequalities for function spaces on cellular domains},
author = {Benjamin Scharf},
journal= {arXiv preprint arXiv:1306.3131},
year = {2013}
}