Trace operators in Besov and Triebel-Lizorkin spaces
Let \({\mathfrak B}^s_{p,q} (\mathbb R^n)\) and \({\mathfrak F}^s_{p,q} (\mathbb R^n)\) be \(B\)-spaces and \(F\)-spaces defined for all \(0<p,q \leq \infty\), \(s>0\), in terms of atomic representations (without moment conditions). These \(B\)-spaces coincide with the classical \(B\)-spaces defined in terms of differences. This applies also to \(F\)-spaces under some restrictions. The paper deals with the traces of these spaces on the hyperplane \(\mathbb R^{n-1} = \{ x \in \mathbb R^n: x_n =0 \}\) and iteratively on \(\mathbb R^m\), \(m<n\). The main result of the paper says that \[ \operatorname{Tr}{\mathfrak B}^s_{p,q} (\mathbb R^n) = {\mathfrak B}^{s - \frac{1}{p}}_{p,q} (\mathbb R^{n-1}) \] if \(0<p,q \leq \infty\), \(s >1/p\). There are corresponding assertions for the limiting case \(s = 1/p\) (with \(L_p (\mathbb R^{n-1})\) as the trace space) and for \({\mathfrak F}^s_{p,q} (\mathbb R^n)\). If \(s < 1/p\) then, \[ \{ g \in D(\mathbb R^n): \;g| \mathbb R^{n-1} = 0 \} \] is dense in \({\mathfrak B}^s_{p,q} (\mathbb R^n)\), \(p<\infty\), \(q<\infty\). This dichotomy problem (either trace or indicated density) is studied in detail with respect to the hyperplanes \(\mathbb R^m\), \(m<n\), and in the limiting situations \(s = \frac{n-m}{p}\).
scientific article; zbMATH DE number 5770968