Restriction quotient and Bochner interpolation for the exterior extension

Let $Y_j$ be the whole-space endpoints $L^q({\mathbb{R}}^n),W^{1,q}({\mathbb{R}}^n)$. Apply the strip operator theorem to $P:Y_j\to X_j$ and the same $E:X_j\to Y_j$. The whole-space Bessel input identifies $Y_s=[Y_0,Y_1{]}_s$ with $H^{s,q}$, so $P:Y_s\to X_s$ and $E:X_s\to Y_s$ are bounded, $PE=I$, and $P{Y}_s=X_s$. This proves the set equality with restriction classes rather than choosing an independent fractional extension.

For a bounded linear spatial map $A:Z\to Z\mathrm{\prime }$ between the Banach spaces in question and a strongly measurable $f$, F6 supplies measurable simple approximants converging almost everywhere in norm. Their images under $A$ are measurable simple functions converging to $Af$. Thus $Af$ is strongly measurable, independent of null-set representatives, and $\mathrm{\parallel }Af{\mathrm{\parallel }}_{L^p(Z\mathrm{\prime })}\le \mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(Z)}$. This applies to the common endpoint maps $E,P$ and to the spatial inclusions $X_1\hookrightarrow X_0$, $Y_1\hookrightarrow Y_0$.

Apply F7 to the realification of each $X_j,Y_j$. Its simple functions, null-set classes, norm and metric are unchanged by realification, so its strong measurability and Bochner $L^p$ space are the same sets with the same norm as for complex values. The real completeness theorem therefore proves metric completeness of the complex Bochner spaces. Complex scalar multiplication on classes obeys $\mathrm{\parallel }\lambda f{\mathrm{\parallel }}_{L^p}=\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p}$, directly from the spatial norm and integral, so these are complex Banach spaces. AC supplies the theorem's Countable Choice hypothesis.

In the interpolation norm let $C_P,C_E$ be these bounds, including the Bessel norm equivalences. If $Pf=u$, then $\mathrm{\parallel }u{\mathrm{\parallel }}_{X_s}\le C_P\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{s,q}}$. Conversely $Eu$ extends $u$ and $\mathrm{\parallel }Eu{\mathrm{\parallel }}_{H^{s,q}}\le C_E\mathrm{\parallel }u{\mathrm{\parallel }}_{X_s}$. Taking the infimum over extensions gives $C_P^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{X_s}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}}\le C_E\mathrm{\parallel }u{\mathrm{\parallel }}_{X_s}$. Since $X_s$ continuously injects into $X_0=L^q(\mathrm{\Omega })$, $P:H^{s,q}\to L^q(\mathrm{\Omega })$ is continuous. Its kernel is closed. The quotient theorem makes the quotient norm definite and complete, proving its well-definedness.

The spatial inequalities $\mathrm{\parallel }u{\mathrm{\parallel }}_{X_0}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{X_1}$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_{Y_0}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{Y_1}$ integrate to continuous injective embeddings $L^p(0,T;X_1)\hookrightarrow L^p(0,T;X_0)$ and $L^p(0,T;Y_1)\hookrightarrow L^p(0,T;Y_0)$. Injectivity follows because zero in the larger spatial space is zero in the smaller one at each time outside a null set. Each pair is consequently a compatible complex Banach couple in its zeroth endpoint, a Hausdorff normed space. The time-valued $E,P$ are single complex-linear maps on these endpoint sums, with the endpoint bounds from step 1.2 and $PE=I$ inherited almost everywhere from the spatial identity. The strip theorem therefore applies. Example 2.7(i), with the same finite exponent $p$ at both endpoints on $(0,T)$, identifies the two interpolated time spaces with $L^p(0,T;X_s)$ and $L^p(0,T;Y_s)$. Together with step 1.1 this gives the displayed Bochner identity and the claimed bounded retraction maps, including $PE=I$, on the interpolated time spaces.