The H1 trace and lifting theorem on a compact Lipschitz boundary

The flat trace estimate. First take $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ and write $\widehat{u}(\xi ,\tau )$ for its full Fourier transform, with $\xi \in {\mathbb{R}}^2$ tangential and $\tau \in \mathbb{R}$ normal. Fourier inversion [F2] in $\tau$ and Cauchy–Schwarz [F4] give $\mathrm{\mid }\widehat{u(\cdot ,0)}(\xi ){\mathrm{\mid }}^2\le C({\int }_{\mathbb{R}}(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2+{\tau }^2)\mathrm{\mid }\widehat{u}(\xi ,\tau ){\mathrm{\mid }}^{2}d\tau )(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{-1\mathrm{/}2},$ because ${\int }_{\mathbb{R}}(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2+{\tau }^2{)}^{-1}d\tau =\pi (1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$. Multiplying by $(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, integrating in $\xi$, and applying [F1] proves $\mathrm{\parallel }u(\cdot ,0){\mathrm{\parallel }}_{H^{1\mathrm{/}2}({\mathbb{R}}^2)}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1({\mathbb{R}}^3)}$.

A flat right inverse. Given $g\in H^{1\mathrm{/}2}({\mathbb{R}}^2)$, put $a(\xi )=(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ and define $\widehat{Eg}(\xi ,t)=\widehat{g}(\xi )e^{-a(\xi )\mathrm{\mid }t\mathrm{\mid }}\chi (t)$, where $\chi \in C_c^{\mathrm{\infty }}(\mathbb{R})$ equals $1$ near $0$. Then $Eg(\cdot ,0)=g$. Direct integration uses ${\int }_0^{\mathrm{\infty }}{e}^{-2at}dt=(2a{)}^{-1}$ and $\mathrm{\mid }\xi {\mathrm{\mid }}^2\le a^2$ to give $\mathrm{\parallel }Eg{\mathrm{\parallel }}_{H^1({\mathbb{R}}^3)}^2\le C{\int }_{{\mathbb{R}}^2}(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\mathrm{\mid }\widehat{g}(\xi ){\mathrm{\mid }}^{2}d\xi$. Plancherel [F1] therefore gives a bounded right inverse on the flat boundary.

Flattening an arbitrary Lipschitz chart. Extend its graph function to a globally Lipschitz function with the same constant by the infimum extension, and use the shear in [F7]. The whole-space extension and smooth density in [F92], followed by bounded shear pullback, reduce the one-sided trace estimate to step 1.1; the endpoint trace of [F7] identifies the continuous boundary values on the smooth approximants and hence the one-sided trace independently of an extension. For the fixed Lipschitz chart partition, multiplication by a bounded Lipschitz cutoff $\chi$ is bounded on $H^{1\mathrm{/}2}({\mathbb{R}}^2)$: split the difference of $\chi g$ into $\chi (x)(g(x)-g(y))+g(y)(\chi (x)-\chi (y))$ and use $\mathrm{\mid }\chi (x)-\chi (y)\mathrm{\mid }\le \min (L_{\chi }\mathrm{\mid }x-y\mathrm{\mid },2\mathrm{\parallel }\chi {\mathrm{\parallel }}_{\mathrm{\infty }})$. In two dimensions its extra Slobodeckij integral is bounded by $C\mathrm{\parallel }g{\mathrm{\parallel }}_2^2$, since the radial integrals near zero and infinity are respectively ${\int }_0^{1}dr$ and ${\int }_1^{\mathrm{\infty }}{r}^{-2}dr$. Thus [F8] gives the required Bessel multiplier bound. On chart overlaps, [F7] supplies the same intrinsic surface measure; graph distances are bi-Lipschitz comparable and its graph area densities are bounded above and below. Consequently the localized surface double integrals, with denominator $\mathrm{\mid }x-y{\mathrm{\mid }}^3$, compare in both directions with their planar pullbacks using the graph measure in each variable, without any unproved Lipschitz volume substitution. Positive support margins inside each chart bound all zero-extension tail terms by $C\mathrm{\parallel }g{\mathrm{\parallel }}_2^2$. These observations justify the finite chart norm and the summed trace estimate. To lift $g$, take each zero-extended localized input $({\chi }_{j}g)\circ {\mathrm{\Phi }}_j$ in that Bessel space, use the flat right inverse of step 2.1, and push it through the inverse shear of [F7]. Multiply by a smooth ambient cutoff supported in the chart and equal to one near the support of ${\chi }_j$ on the surface. Its trace is exactly ${\chi }_{j}g$, not ${\chi }_j^{2}g$. The finite sum therefore has trace ${\sum }_j{\chi }_{j}g=g$ and bounded whole-space $H^1$ norm. The chart cutoffs give bounded support near $\mathrm{\Gamma }$. The same construction on the other side proves the localized exterior-collar assertion.

The bounded extension $E:H^1({\mathrm{\Omega }}^{-})\to H^1({\mathbb{R}}^3)$ with the correct restriction is supplied directly by the already proved general Lipschitz extension construction in [F92]; that construction assumes no boundary trace theorem. Its compact-support cutoff is allowed there. In a flattened collar, the one-dimensional fundamental theorem and Cauchy--Schwarz give $\mathrm{\parallel }f(\cdot ,t)-\tau f{\mathrm{\parallel }}_{L^2(K)}^2\le t{\int }_0^t\mathrm{\parallel }{\mathrm{\partial }}_{s}f(\cdot ,s){\mathrm{\parallel }}_{L^2(K)}^{2}ds$ for a compact tangential set $K$ and $t>0$, with $t$ measured into the chosen side. The weak-coordinate and endpoint argument of [F7] passes this to $H^1$ functions. Thus the trace just constructed agrees with the boundary restriction for a continuous $H^1$ representative, since continuity gives the same local $L^2$ limit.

The homogeneous energy trace. For $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$, [F3] gives $\mathrm{\parallel }u{\mathrm{\parallel }}_6\le C\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2$; hence it continues to every element of ${\dot{H}}^1$ by completion. Fix $\zeta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ equal to $1$ near $\mathrm{\Gamma }$. Hölder's inequality [F5] on the bounded support of $\mathrm{\nabla }\zeta$ gives $\mathrm{\parallel }\zeta u{\mathrm{\parallel }}_{H^1}\le C(\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2+\mathrm{\parallel }u{\mathrm{\parallel }}_6)\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{{\dot{H}}^1}$. Applying the trace bound of step 3.1 to $\zeta u$ proves $\mathrm{\parallel }\gamma u{\mathrm{\parallel }}_{H^{1\mathrm{/}2}(\mathrm{\Gamma })}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{{\dot{H}}^1}$; the compactly supported right inverse from step 3.1 is also a right inverse into ${\dot{H}}^1$.