The half-space trace lies in the fractional Slobodeckij space

The estimate for smooth compactly supported $u$. Let $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ and let $g=u(\cdot ,0)$ be its classical boundary value. Fix $i\in \{1,\dots ,d\}$, $h>0$, put $t:=h\mathrm{/}2$, and fix $x^{\mathrm{\prime }}\in {\mathbb{R}}^d$. Splitting the increment at the midpoint and applying the fundamental theorem of calculus along the vertical and tangential segments gives $\mathrm{\mid }g(x^{\mathrm{\prime }}+h{e}_i)-g(x^{\mathrm{\prime }})\mathrm{\mid }\le {\int }_0^t\mathrm{\mid }{\mathrm{\partial }}_{n}u(x^{\mathrm{\prime }},s)\mathrm{\mid }ds+{\int }_0^t\mathrm{\mid }{\mathrm{\partial }}_{n}u(x^{\mathrm{\prime }}+h{e}_i,s)\mathrm{\mid }ds+h{\int }_0^1\mathrm{\mid }{\mathrm{\partial }}_{i}u(x^{\mathrm{\prime }}+sh{e}_i,t)\mathrm{\mid }ds$; raising to the $p$-th power, integrating in $x^{\mathrm{\prime }}$ and using translation invariance of Lebesgue measure makes the two normal-line integrals equiponderant, so ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }g(x^{\mathrm{\prime }}+h{e}_i)-g(x^{\mathrm{\prime }}){\mathrm{\mid }}^{p}d{x}^{\mathrm{\prime }}\le 3^{p-1}(2\int A_i(x^{\mathrm{\prime }},h{)}^{p}d{x}^{\mathrm{\prime }}+\int (h{\int }_0^1\mathrm{\mid }{\mathrm{\partial }}_{i}u(x^{\mathrm{\prime }}+sh{e}_i,t)\mathrm{\mid }ds{)}^{p}d{x}^{\mathrm{\prime }})$ with $A_i(x^{\mathrm{\prime }},h):={\int }_0^{h\mathrm{/}2}\mathrm{\mid }{\mathrm{\partial }}_{n}u(x^{\mathrm{\prime }},s)\mathrm{\mid }ds$. Multiplying by $h^{-p}$ and integrating in $h$, Hardy's inequality [F2] applied in the normal variable (with Tonelli [F7]) bounds the first term by $2^{2-p}(p^{\mathrm{\prime }}{)}^p\mathrm{\parallel }{\mathrm{\partial }}_{n}u{\mathrm{\parallel }}_{L^p(H)}^p$, while Holder [F6] applied to the inner $s$-integral followed by Tonelli, the substitution $t=h\mathrm{/}2$ and translation invariance bounds the second term by $2\mathrm{\parallel }{\mathrm{\partial }}_{i}u{\mathrm{\parallel }}_{L^p(H)}^p$. Summing over $i=1,\dots ,d$ and using the coordinate-direction form [F1] of the seminorm gives $[g{]}_{\theta ,p}^p\le C(d,p){\sum }_{j=1}^{d+1}\mathrm{\parallel }{D}_{j}u{\mathrm{\parallel }}_{L^p(H)}^p=C(d,p){\int }_H\mathrm{\mid }Du{\mathrm{\mid }}^p$.

Scale invariance of the homogeneous estimate. For $r>0$ and measurable $u$ on $H$ put $u_r(x^{\mathrm{\prime }},t):=u(r{x}^{\mathrm{\prime }},rt)$, so $g_{u_r}(x^{\mathrm{\prime }})=g(r{x}^{\mathrm{\prime }})$ and $D_j{u}_r=r(D_{j}u)(r\cdot )$ for $j=1,\dots ,d+1$. The change of variables $\xi =r{x}^{\mathrm{\prime }}$, $\eta =r{y}^{\mathrm{\prime }}$ gives $[g_{u_r}{]}_{\theta ,p}^p=\int \int \mathrm{\mid }g(\xi )-g(\eta ){\mathrm{\mid }}^p\mid \frac{\xi -\eta }{r}{\mid }^{-d-p\theta }{r}^{-2d}d\xi d\eta =r^{-d+p\theta }[g{]}_{\theta ,p}^p$, and the change of variables $(x^{\mathrm{\prime }},t)\mapsto (r{x}^{\mathrm{\prime }},rt)$ gives ${\int }_H\mathrm{\mid }D{u}_r{\mathrm{\mid }}^{p}dx=r^p\cdot r^{-d-1}{\int }_H\mathrm{\mid }Du{\mathrm{\mid }}^{p}dx=r^{-d+p\theta }{\int }_H\mathrm{\mid }Du{\mathrm{\mid }}^{p}dx$ because $r^p{r}^{-d-1}=r^{-d+p-1}$ and $p-1=p\theta$. Hence both sides of the homogeneous estimate carry the same scaling exponent.

The general case by density and Fatou. Let $u\in W^{1,p}(H)$ and let ${\phi }_m\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ be such that $u_m:={\phi }_m{\mathrm{\mid }}_H\to u$ in $W^{1,p}(H)$; such a sequence exists by [F4]. Step 1.1 applies to each $u_m$, giving $[T_{+}{u}_m{]}_{\theta ,p}^p\le C(d,p){\int }_H\mathrm{\mid }D{u}_m{\mathrm{\mid }}^p$. By [F3] $T_{+}{u}_m\to T_{+}u$ in $L^p({\mathbb{R}}^d)$, so a subsequence converges almost everywhere; Fatou's lemma [F5] applied to the nonnegative integrands of the seminorm gives $[T_{+}u{]}_{\theta ,p}^p\le {\mathrm{lim\hspace{0.17em}inf}}_m[T_{+}{u}_m{]}_{\theta ,p}^p$, while ${\int }_H\mathrm{\mid }D{u}_m{\mathrm{\mid }}^p\to {\int }_H\mathrm{\mid }Du{\mathrm{\mid }}^p$ by the norm convergence. Hence $[T_{+}u{]}_{\theta ,p}^p\le C(d,p){\int }_H\mathrm{\mid }Du{\mathrm{\mid }}^p\le C(d,p)\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(H)}^p$ for every $u\in W^{1,p}(H)$, and $T_{+}$ is bounded into $W^{\theta ,p}({\mathbb{R}}^d)$.

Conclusion. Step 1.1 proves the homogeneous bound for the dense smooth class; step 2.1 extends it to all of $W^{1,p}(H)$ by continuity of the trace and Fatou, giving the displayed chain and the boundedness of $T_{+}:W^{1,p}(H)\to W^{\theta ,p}({\mathbb{R}}^d)$; step 1.2 verifies the scaling of both sides of the homogeneous estimate.