Traces on fractals of function spaces with Muckenhoupt weights (Q2466276)

Let \(\Gamma\) be a compact \(d\)-set in \(\mathbb R^n\) with \(0<d<n\) and let \(w^\Gamma_\kappa (x)\) be a positive weight function in \(\mathbb R^n\) with \(w^\Gamma_\kappa (x) = \text{dist}(x, \Gamma)^\kappa\) near \(\Gamma\). The author studies traces \(\text{tr}_\Gamma\) of weighted spaces \(B^s_{pq} (\mathbb R^n, w^\Gamma_\kappa)\) and \(F^s_{pq} (\mathbb R^n, w^\Gamma_\kappa)\) on \(\Gamma\). By Theorem 3.1 one has \[ \text{ tr}_\Gamma B^{\frac{\kappa}{p} + \frac{n-d}{p}}_{pq} (\mathbb R^n, w^\Gamma_\kappa ) = L_p (\Gamma), \] where \(\kappa > -n+d\), \(0<p<\infty\), \( 0<q \leq \min (1,p)\). Let \(B^s_{pq} (\Gamma)\) be Besov spaces on \(\Gamma\). Then \[ \text{tr}_\Gamma \, B^s_{pq} (\mathbb R^n, w^\Gamma_\kappa ) = B^{s- \frac{n-d}{p} - \frac{\kappa}{p}}_{pq} (\Gamma), \] where \(-(n-d) < \kappa < sp - (n-d)\). There are corresponding assertions for the spaces \(F^s_{pq} (\mathbb R^n, w^\Gamma_\kappa)\).