Restriction-quotient interpolation from a common endpoint retraction

For a whole-space $H^2$ extension $F$ of $u$, [F4] identifies $F$ with its complex weak $W^{2,2}({\mathbb{R}}^n)$ class. The contraction in [F8] then gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(D)}\le \mathrm{\parallel }F{\mathrm{\parallel }}_{W^{2,2}({\mathbb{R}}^n)}\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{H^2}$. Conversely the assumed endpoint extension sends every weak $u\in W^{2,2}(D)$ to $Eu\in H^2({\mathbb{R}}^n)$ with norm at most $C\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{2,2}(D)}$. Taking infima proves $W^{2,2}(D)=H^2(D)$ and the norm equivalence. The order-zero quotient is exactly $L^2(D)$ by zero extension. Both domain endpoints inject continuously into $L^2(D)$, so form a compatible Banach couple.

On the nonzero complex Hilbert space $K=L^2({\mathbb{R}}_{\xi }^n)$, let $M=M_{1+\mathrm{\mid }\xi {\mathrm{\mid }}^2}$. Lebesgue measure is sigma-finite and this real multiplier is finite and at least one, so [F6] gives a self-adjoint $M\ge I$ with $D(M^{t\mathrm{/}2})=\{g\in K:⟨\xi {⟩}^{t}g\in K\}$ and norm $\mathrm{\parallel }⟨\xi {⟩}^{t}g{\mathrm{\parallel }}_2$ for $t\ge 0$. The Fourier identifications in [F6] and [F7] make ${\mathcal{F}}_2$ an onto isometry from $H^t({\mathbb{R}}^n)$ to this domain for $0\le t\le 2$, using actual $L^2$ representatives at these nonnegative orders. At $t=0,2$ these Fourier maps and their inverse are single consistent maps on the endpoint sums. Thus [F3] transports them to inverse contractions on the interpolation spaces. Apply [F5] to $M$ with scale indices $0,2$: its interpolation space is $D(M^{\theta })$ with exactly the weighted norm of order $2\theta$. Transporting back by the same Fourier maps yields $[L^2({\mathbb{R}}^n),H^2({\mathbb{R}}^n){]}_{\theta }=H^{2\theta }({\mathbb{R}}^n)$ isometrically in the bracket norms for $0<\theta <1$. This is an application of the spectral supplier's stated interpolation contract, rather than an extra conclusion attributed to [F2].

Put $X_{\theta }=[L^2(D),H^2(D){]}_{\theta }$ and $Y_{\theta }=[L^2({\mathbb{R}}^n),H^2({\mathbb{R}}^n){]}_{\theta }=H^{2\theta }({\mathbb{R}}^n)$. The whole-space identity is supplied by step 1.2. At the two endpoints $r_D$ and $E$ are bounded common maps and $r_{D}E=I$. Applying them to a strip function preserves bounded sum-continuity, norm-holomorphy, endpoint continuity and decay by boundedness and linearity; hence the strip-map theorem gives bounded $r_D:Y_{\theta }\to X_{\theta }$ and $E:X_{\theta }\to Y_{\theta }$ with the same identity. In particular $X_{\theta }=r_D{Y}_{\theta }$ as sets. This conclusion uses actual common maps, not unrelated endpoint extensions.

If $u=r_{D}F$ with $F\in Y_{\theta }$, the interpolated restriction bound gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{X_{\theta }}\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{H^{2\theta }}$, so $\mathrm{\parallel }u{\mathrm{\parallel }}_{X_{\theta }}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{2\theta }(D)}$ after infimizing. In the other direction, $Eu$ is an extension with $\mathrm{\parallel }Eu{\mathrm{\parallel }}_{H^{2\theta }}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{X_{\theta }}$; therefore $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{2\theta }(D)}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{X_{\theta }}$. This proves equality with equivalent norms and the retraction at every intermediate order. Changing the second endpoint to its equivalent weak norm changes the strip norm only by bounded identity maps in both directions, so gives the same space with equivalent norms. At $s=0,2$ these are the specified endpoint statements. No zero trace is present in either endpoint or quotient construction.