Atomic decompositions of function spaces with Muckenhoupt weights; an example from fractal geometry

Preprint series: 05-17, Analysis

The paper is published: Jenaer Schriften zur Mathematik und Informatik, Math/Inf/06/05, 2005

MSC:
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
Abstract: This paper deals with atomic decompositions in spaces of type $B^s_{p,q}(\rn, w)$, $F^s_{p,q}\rn, w)$, $0<p<\infty$, $0<q\leq\infty$, $s\in\real$, where the weight function $w$ belongs to some Muckenhoupt class $\mathcal{A}_r$. In particular, we consider the weight function $w_\varkappa (x) = \dist(x,\Gamma)^\varkappa$, where $\Gamma$ is some $d$-set, $0<d<n$, and $\varkappa > -(n-d)$.
Keywords: weighted function spaces, atomic decomposition, Muckenhoupt weights, $d$-sets