arXiv subjects
Publications and source records attributed to Aleksei Y. Chikalov.
Let $d\in [0, n]$ and let $E\subset \mathbb{R}^n$ be a closed $d$-thick set. Given $p\in [1, \infty]$, $q\in (0, \infty]$, and $s\in (\frac{n-d}{p}, 1)$, we provide an intrinsic description of the trace-space of the Besov space $B^s_{p, q}(\mathbb{R}^n)$ to $E$. If, in addition, $p<\infty$, we give an intrinsic description of the trace-space of the Lizorkin--Triebel space $F^s_{p, q}(\mathbb{R}^n)$ to $E$.
Let $p\in[1,\infty)$, and let $X=(X,d,μ)$ be a metric measure space such that $μ$ is uniformly locally doubling and $X$ supports a local weak $(1,p)$-Poincaré inequality. Given $θ\in(0,p)$ and an Ahlfors--David codimension-$θ$ regular set $E\subset X$, the trace-space of the Besov space $B^{θ/p}_{p,1}(X)$ to $E$ can be identified with $L_p(E,\mathcal{H}_θ\lfloor_E)$. If there exists a measurable set $A\subset E$ with $0<\mathcal H_θ(A)<\infty$ such that $\mathcal H_θ\lfloor_A$ is nonatomic, we prove that there is no bounded linear extension operator $\operatorname{Ext}:L_p(E,\mathcal{H}_θ\lfloor_E) \to B^{θ/p}_{p,1}(X)$.
Given $n\in \mathbb{N}$, $p\in [1,\infty)$, and a weight $γ$ satisfying the local Muckenhoupt $A_p$ condition, we introduce a weakened version of the Ahlfors--David codimension-$θ$ regularity condition for Ahlfors--David $d$-regular sets $E\subset\mathbb{R}^n$, where $d\in(0,n)$ and $θ\in(0,p)$. Under this assumption, we provide a complete intrinsic description of the trace-space of the weighted Besov space $B^{\fracθ{p}}_{p,1}(\mathbb{R}^n,γ)$ to $E$. In particular, our results cover the case of power-type weights $γ(x)=|x|^α$ with $-n<α<n(p-1)$, $α\neq -(n-1)$, when $E=\mathbb{R}^{n-1}$. This extends earlier results obtained by Haroske and Schmeisser.
Let $(X,d,μ)$ be a metric measure space whose measure $μ$ is uniformly locally doubling and which supports a local weak $(1,p)$-Poincaré inequality for some $p\in[1,\infty)$. Given $θ\in(0,p)$ and an Ahlfors--David codimension-$θ$ regular subset $E\subset X$, we provide a complete intrinsic description of the trace-space of the Besov space $B^{θ/p}_{p,1}(X)$ to $E$. More precisely, we show that the trace operator is well defined and bounded from $B^{θ/p}_{p,1}(X)$ to $L_p(E,\mathcal H_θ\lfloor_E)$. We also show that the upper estimate in the Ahlfors--David codimension-$θ$ regularity condition is necessary for such boundedness under the local weak Poincaré inequality. Conversely, assuming that $E$ is Ahlfors--David codimension-$θ$ regular, we construct a bounded nonlinear extension operator from $L_p(E,\mathcal H_θ\lfloor_E)$ to $B^{θ/p}_{p,1}(X)$. Thus the trace-space is identified intrinsically with $L_p(E,\mathcal H_θ\lfloor_E)$. This extends the classical limiting case of the trace theorem obtained by Burenkov and Gol'dman. Finally, we apply the general theory to $K$-regular trees, $K\ge 1$, for which we additionally derive a necessary and sufficient criterion for the existence of traces.