Characterisation of function spaces via mollification; fractal quantities for distributions
The aim of the paper is twofold. The first one is a characterization of Besov spaces in terms of mollifications. The idea is based on the multifractal quantity \[ \mu^\lambda_{p,q}=\biggl(\sum_j 2^{j\lambda q} \biggl(\sum_m \mu(Q_{j,m})^p\biggr)^{q/p}\biggr)^{1/q} \] introduced for Radon measures \(\mu\). Here \(Q_{j,m}\) denotes a standard dyadic cube. It is known that \(\mu\in B^s_{p,q}({\mathbb R}^n)\), \(s<0\), iff \(\mu^\lambda_{p,q}<\infty\) and \(s=\lambda+n(\frac 1 p -1)\). The aim of the paper is to extend this characterization to positive smoothnesses, \(s>0\). For this purpose, mollifications are used. The second aim is to study multifractal quantities for distributions. In particular, generalizations \(\lambda^f\), \(s_f\), \(f\in {\mathcal S}'({\mathbb R}^n)\), of the quantities \(\lambda_\mu(\frac 1 p)=\sup\{\lambda:\, \mu^\lambda_{p,q}<\infty\}\) and \(s_\mu(\frac 1 p)=\sup\{s: \mu\in B^s_{p,q}({\mathbb R}^n)\}\) are considered. It is proven that \(s_f(\frac 1 p)= \lambda^f(\frac 1 p)+ n(\frac 1 p - 1)\) if \(f\) has compact support.
scientific article; zbMATH DE number 2111355