Fractional Sobolev regularity on a simplex

Assume Countable Choice. Let $U\subset {\mathbb{R}}^d$ be a bounded open set, $0<r<1$, and let $q$ be a measurable real function on $U$. Define $[q{]}_{H^r(U)}^2:={\int }_U{\int }_U\frac{\mathrm{\mid }q(x)-q(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d+2r}}dxdy,$ with the diagonal integrand set to zero, and $H^r(U):=\{q\in L^2(U):[q{]}_{H^r(U)}<\mathrm{\infty }\}$, using almost-everywhere classes and norm $\mathrm{\parallel }q{\mathrm{\parallel }}_{H^r(U)}^2:=\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(U)}^2+[q{]}_{H^r(U)}^2$. For $u\in H^1(U)$, write $u\in H^{1+r}(U)$ when each weak derivative ${\mathrm{\partial }}_{j}u$ belongs to $H^r(U)$, and define $\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(U)}^2:={\sum }_{j=1}^d[{\mathrm{\partial }}_{j}u{]}_{H^r(U)}^2\mathrm{.}$ For vector fields, use the sum of the componentwise squared $H^r$ norms. Use these definitions on each cell $K$ and on the full domain $\mathrm{\Omega }$, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{1+r}(U)}^2:=\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1(U)}^2+\mathrm{\mid }u{\mathrm{\mid }}_{H^{1+r}(U)}^2$. At $r=1$, set $H^1(U)=W^{1,2}(U)$, interpret $[q{]}_{H^1(U)}^2:=\mathrm{\parallel }\mathrm{\nabla }q{\mathrm{\parallel }}_{L^2(U)}^2$, and set $H^{1+r}(U)=W^{2,2}(U)$ with the integer-order membership of Integer-order Sobolev spaces and their norms, retaining the displayed full-norm rule. Thus $\mathrm{\mid }u{\mathrm{\mid }}_{H^2(U)}^2={\sum }_{j,k}\mathrm{\parallel }{\mathrm{\partial }}_k{\mathrm{\partial }}_{j}u{\mathrm{\parallel }}_{L^2(U)}^2$. This Hessian convention counts mixed derivatives twice: its full norm is equivalent to, rather than identical with, the ordinary multi-index $W^{2,2}$ norm, with $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(U)}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^2(U)}\le \sqrt{2}\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(U)}$. The local seminorm is the restriction of the Slobodeckij convention in The Gagliardo--Slobodeckij space on Euclidean space to $U\times U$.