The sharp trace theorem: boundedness and range in the fractional space

Boundedness in the fractional norm. Let $u\in W^{1,p}(\mathrm{\Omega })$. By [F1], $Tu={\sum }_{j}T({\chi }_{j}u)$, and each ${\chi }_{j}u$ is supported in one chart. Flattening the $j$-th piece and reflecting gives $w_j\in W^{1,p}(H)$ with $\mathrm{\parallel }{w}_j{\mathrm{\parallel }}_{W^{1,p}(H)}\le C_j\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$ by [F5], whose flat trace is the chart representation $({\chi }_{j}Tu)\circ {\mathrm{\Psi }}_j^{-1}$ of the $j$-th summand; [F2] bounds its $W^{\theta ,p}({\mathbb{R}}^{n-1})$ norm by $C_j^{\mathrm{\prime }}\mathrm{\parallel }{w}_j{\mathrm{\parallel }}_{W^{1,p}(H)}$. Each chart $L^p$ term is already bounded by the same local half-space bound; their finite sum is controlled by $C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$. Adding the finitely many seminorm bounds and using the atlas-independence [F3] to pass to the norm of [F6] gives $\mathrm{\parallel }Tu{\mathrm{\parallel }}_{W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })}\le C(\mathrm{\Omega },p)\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$.

Surjectivity. Let $g\in W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })$. For each $j$ let $g_j:=({\chi }_{j}g)\circ {\mathrm{\Psi }}_j^{-1}\in W^{\theta ,p}({\mathbb{R}}^{n-1})$ be the localised chart representation, and let $w_j:=R_{+}{g}_j\in W^{1,p}(H)$ be its flat lift, so that $T_{+}{w}_j=g_j$ and $\mathrm{\parallel }{w}_j\mathrm{\parallel }\le C_j\mathrm{\parallel }{g}_j\mathrm{\parallel }$ by [F4]. Pulling $w_j$ back through the chart and multiplying by a smooth cutoff supported in the chart and equal to $1$ near $\mathrm{supp}{\chi }_j$ gives $u_j\in W^{1,p}(\mathrm{\Omega })$ with $\mathrm{\parallel }{u}_j\mathrm{\parallel }\le C_j^{\mathrm{\prime }}\mathrm{\parallel }{g}_j\mathrm{\parallel }$ by [F5] and, by the chart-transport part of [F1], $T{u}_j=({\chi }_{j}g){\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$. Setting $u:={\sum }_j{u}_j$ gives $Tu={\sum }_j{\chi }_{j}g=g$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}\le C(\mathrm{\Omega },p)\mathrm{\parallel }g{\mathrm{\parallel }}_{W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })}$ (using [F3] to compare the two atlas expressions and the triangle inequality). Hence $T$ is onto.

Strictness of the range in $L^p$. Put $a=1\mathrm{/}p-\min (\theta ,1\mathrm{/}p)\mathrm{/}2$, so $0<a<1\mathrm{/}p$ and $p(a+\theta )>1$. Choose a chart and a smooth cutoff $\beta$ supported inside it and equal to one on a small coordinate box centred at zero. Set $q(y)=\beta (y)\mathrm{\mid }{y}_1{\mathrm{\mid }}^{-a}$ off $y_1=0$, and zero on that null hyperplane. Since $ap<1$, $q\in L^p({\mathbb{R}}^{n-1})$. For small $h>0$, restrict $y_1$ to $(h,2h)$ and the remaining coordinates to a fixed smaller box, where both cutoffs are one. Then $\mathrm{\mid }q(y+h{e}_1)-q(y)\mathrm{\mid }\ge c_a{h}^{-a}$, and $\int \mathrm{\mid }q(y+h{e}_1)-q(y){\mathrm{\mid }}^{p}dy\ge c{h}^{1-ap}$. By [F8], $[q{]}_{\theta ,p}^p\ge c^{\mathrm{\prime }}{\int }_0^{h_0}{h}^{-p}{h}^{1-ap}dh=c^{\mathrm{\prime }}{\int }_0^{h_0}{h}^{-p(a+\theta )}dh=+\mathrm{\infty }$. Transport $q$ to the boundary and extend by zero. The bounded positive chart density gives boundary $L^p$ membership. Choose a subordinate atlas cutoff equal to one on this support; its local norm is infinite, so [F3] gives nonmembership in the boundary fractional space for every atlas. Thus the trace range is a strict subset of $L^p$.

Non-compactness. In one boundary chart choose a nonzero $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$, $d=n-1$, supported near its centre, and a smooth normal cutoff $\chi$ equal to one near zero. Let $C_m=\mathrm{\parallel }\psi (m\cdot ){\mathrm{\parallel }}_{W^{\theta ,p}}=m^{-d\mathrm{/}p}\mathrm{\parallel }\psi {\mathrm{\parallel }}_p+m^{\theta -d\mathrm{/}p}[\psi {]}_{\theta ,p}$ and $g_m=\psi (m\cdot )\mathrm{/}{C}_m$. Here $0<[\psi {]}_{\theta ,p}<\mathrm{\infty }$: finiteness follows from the Lipschitz increment bound near zero and the integrable tail, and positivity follows because $\psi$ is not constant. Set $w_m(y,t)=g_m(y)\chi (mt)$ and pull it back through the reflected chart, multiplying by a fixed ambient cutoff equal to one near the centre. For all sufficiently large $m$, this cutoff is one on the support; call the resulting class $u_m$. Scaling gives $\mathrm{\parallel }{w}_m{\mathrm{\parallel }}_p^p\le C{C}_m^{-p}{m}^{-d-1}$ and ${\sum }_j\mathrm{\parallel }{D}_j{w}_m{\mathrm{\parallel }}_p^p\le C{C}_m^{-p}{m}^{p-d-1}$, so $\mathrm{\parallel }{u}_m{\mathrm{\parallel }}_{W^{1,p}}\le C$ since $p\theta =p-1$. The transported traces $b_m=T{u}_m$ have fractional norms bounded below by $c>0$ by [F3] and [F6], whereas $\mathrm{\parallel }{b}_m{\mathrm{\parallel }}_p\le C{m}^{-d\mathrm{/}p}\mathrm{/}{C}_m\to 0$ because $\theta >0$. If a subsequence converged in the boundary fractional norm, it would converge in boundary $L^p$ to the same limit, necessarily zero. Fractional norm convergence to zero would contradict the lower bound. Hence $T$ is not compact into $W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })$.

Conclusion. Step 1.1 gives the norm bound, step 1.2 gives surjectivity onto $W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })$, step 1.3 shows the range is a strict subset of $L^p(\mathrm{\partial }\mathrm{\Omega })$, and step 1.4 shows the trace is not compact into $W^{\theta ,p}(\mathrm{\partial }\mathrm{\Omega })$; this proves all the assertions of the statement.