Traces of Besov spaces on fractal \(h\)-sets and dichotomy results (Q2787156)

Let \(\bar{\sigma} = (\sigma_j )^\infty_{j=1}\) be a sequence of positive numbers with \(\sigma_j \sim \sigma_{j+1}\) and let \(B^{\bar{\sigma}}_{p,q} (\mathbb R^n)\) be the related generalized Besov spaces covering the classical Besov spaces \(B^s_{p,q} (\mathbb R^n)\) for which \(\sigma_j =2^{js}\), \(s>0\), \(1\leq p<\infty\), \(0<q<\infty\). Let \(B(\gamma, r)\) be balls centered at \(\gamma \in \Gamma\) and of radius \(0<r<1\), where \(\Gamma\) is a compact set in \(\mathbb R^n\) which can be furnished with a Radon measure \(\mu\) on \(\mathbb R^n\) such that \(\text{supp }\mu = \Gamma\) and NEWLINE\[NEWLINE \mu \big( B(\gamma, r) \big) \sim h(r), \qquad r\in (0,1), \quad \gamma \in \Gamma. NEWLINE\]NEWLINE Here, \(h\) is a gauge function: continuous monotonically increasing on \([0,1]\), \(h(0)=0\). Then \(\Gamma\) is called an \(h\)-set, generalizing the well-known \(d\)-sets, \(0<d<n\), where \(h(r) = r^d\). The paper deals with traces of suitable spaces \(B^{\bar{\sigma}}_{p,q} (\mathbb R^n)\) on \(\Gamma\), generating related spaces \(B^{\bar{\lambda}}_{p,q} (\Gamma)\) as subspaces of \(L_p (\Gamma)\). Of special interest is the dichotomy, which means that either \(B^{\bar{\lambda}}_{p,q} (\Gamma) = \text{tr}_\Gamma \, B^{\bar{\sigma}}_{p,q} (\mathbb R^n)\) exists or \(D(\mathbb R^n \setminus \Gamma)\) is dense in \( B^{\bar{\sigma}}_{p,q} (\mathbb R^n)\). Here, \(\text{tr}_\Gamma f\) is the trace of \(f \in B^{\bar{\sigma}}_{p,q} (\mathbb R^n)\), defined as the completion of the pointwise trace of \(\phi \in S(\mathbb R^n)\) on \(\Gamma\).