A trace theorem for Besov functions in spaces of homogeneous type. (English) Zbl 1394.43005

Marcos, Miguel Andrés

Publ. Mat., Barc. 62, No. 1, 185-211 (2018).

A. Jonsson and H. Wallin [Function spaces on subsets of \(\mathbb{R}^n\). Math. Rep. Ser. 2, No. 1, 221 p. (1984; Zbl 0875.46003)] proved that Besov functions on \(\mathbb{R}^n\) leave a Besov trace on certain subsets of dimension \(d < n\). The author proves a similar trace theorem in the metric setting, replacing \(\mathbb{R}^n\) by a space \(X\) of homogeneous type. Certain results are obtained under the assumption that \(X\) is Ahlfors regular. First, the author proves an extension theorem. Namely, given an appropriate set \(F\subset X\), Besov functions on \(F\) are restrictions of more regular Besov functions on \(X\). Next, as an auxiliary result, the author proves that interpolation between certain potential spaces gives Besov spaces. Finally, he obtains a restriction theorem, proving that Besov functions on \(X\) have restrictions on \(F\).

Reviewer: Evgueni Doubtsov (St. Petersburg)

MSC:

43A85

Harmonic analysis on homogeneous spaces

Keywords:

Besov spaces; spaces of homogeneous type; trace theorem; extension theorem; restriction theorem; interpolation

Citations:

Zbl 0875.46003