Equivalent Characterizations and Applications of Fractional Sobolev Spaces with Partially Vanishing Traces on $(ε,δ,D)$-Domains Supporting $D$-Adapted Fractional Hardy Inequalities

Let $Ω\subset\mathbb{R}^n$ be an $(ε,δ,D)$-domain, with $ε\in(0,1]$, $δ\in(0,\infty]$, and $D\subset \partial Ω$ being a closed part of $\partial Ω$, which is a general open connected set when $D=\partial Ω$ and an $(ε,δ)$-domain when $D=\emptyset$. Let $s\in(0,1)$ and $p\in[1,\infty)$. If ${W}^{s,p}(Ω)$, ${\mathcal W}^{s,p}(Ω)$, and $\mathring{W}_D^{s,p}(Ω)$ are the fractional Sobolev spaces on $Ω$ that are defined respectively via the restriction of $W^{s,p}(\mathbb{R}^n)$ to $Ω$, the intrinsic Gagliardo norm, and the completion of all $C^\infty(Ω)$ functions with compact support away from $D$, in this article we prove their equivalences [that is, ${W}^{s,p}(Ω)={\mathcal{W}}^{s,p}(Ω) =\mathring{W}_D^{s,p}(Ω)$] if $Ω$ supports a $D$-adapted fractional Hardy inequality and, moreover, when $sp\ne 1$ such a fractional Hardy inequality is shown to be necessary to guarantee these equivalences under some mild geometric conditions on $Ω$. Using the aforementioned equivalences, we show that the real interpolation space $(L^p(Ω), \mathring{W}_D^{1,p}(Ω))_{s,p}$ equals to some weighted fractional order Sobolev space $\mathcal{W}^{s,p}_{d_D^s}(Ω)$ when $p\in (1,\infty)$. Applying this to the elliptic operator $\mathcal{L}_D$ in $Ω$ with mixed boundary condition, we characterize both the domain of its fractional power and the parabolic maximal regularity of its Cauchy initial problem by means of $\mathcal{W}^{s,p}_{d_D^s}(Ω)$.