**Preprint series:** 06-06, Reports on Analysis

Jan Schneider

**The paper is published:**
Jenaer Schriften zur Mathematik und Informatik, Math/Inf/02/06, 2006

**MSC:**- 46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
- 42C40 Wavelets
- 46B45 Banach sequence spaces [See also 46A45]

**Abstract:** This paper concerns function spaces with varying smoothness and is a self-contained part of [8].We define the spaces $B^{S,s_0}_p(\mathbb{R}^n)$, where the function $S : x \mapsto s(x)$ is negative and determines the smoothness pointwise. First we prove basic properties and then use different wavelet decompositions to get information about local smoothness behavior. The main results are characterizations of the spaces $B^{S,s_0}_p(\mathbb{R}^n)$ by weighted sequence space norms of the wavelet coefficients. These are used to prove an interesting connection to the so-called two-microlocal spaces $C^{s,s'}(x^0)$.

**Keywords:** *Besov spaces, varying smoothness, wavelets, weighted sequence spaces*

**Upload:** 2006-02-15

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