Euclidean complex geometrical optics solvability with an H1 correction

Write $\zeta =a+ib$. The null condition gives $\mathrm{\mid }a\mathrm{\mid }=\mathrm{\mid }b\mathrm{\mid }=:s=\mathrm{\mid }\zeta \mathrm{\mid }\mathrm{/}\sqrt{2}$ and $a\cdot b=0$. Choose an orthogonal map $U$ with $Ua=s{e}_1$ and $Ub=s{e}_2$. Work temporarily with $U\mathrm{\Omega }$, $q\circ U^{-1}$, and the phase $U\zeta$, relabeling them $\mathrm{\Omega }$, $q$, and $\zeta$ for the construction; the Euclidean Laplacian, Lebesgue norms, and null phase are invariant under this change, and the constructed remainder is pulled back by $U$ at the end. As $\mathrm{\Omega }\subset B(0,R)$, choose $L>\max \{1,(R+1)\mathrm{/}\pi \}$; then $\mathrm{\Omega }\subset Q_L=(-\pi L,\pi L{)}^n$. Put ${\eta }_k=(k+\frac{1}{2}{e}_2)\mathrm{/}L$ and $w_k(x)=(2\pi L{)}^{-n\mathrm{/}2}{e}^{i{\eta }_k\cdot x}$ for $k\in {\mathbb{Z}}^n$. By [F1], these form an orthonormal basis on $Q_L$: they are the rescaled standard Fourier characters multiplied by the unitary factor $e^{i{x}_2\mathrm{/}(2L)}$. For $A_{\zeta }=D^2+2\zeta \cdot D$, one has $A_{\zeta }{w}_k=p_k{w}_k$, where $p_k=\mathrm{\mid }{\eta }_k{\mathrm{\mid }}^2+2\zeta \cdot {\eta }_k$ and $\mathrm{\mid }\mathrm{Im}{p}_k\mathrm{\mid }=2s\mathrm{\mid }{k}_2+\frac{1}{2}\mathrm{\mid }\mathrm{/}L\ge s\mathrm{/}L$. If $\mathrm{\mid }{\eta }_k\mathrm{\mid }\le 4s$, then $\mathrm{\mid }{p}_k\mathrm{\mid }\ge s\mathrm{/}L$ and $\mathrm{\mid }{\eta }_k\mathrm{/}{p}_k\mathrm{\mid }\le 4L$. If $\mathrm{\mid }{\eta }_k\mathrm{\mid }>4s$, then $\mathrm{\mid }\mathrm{Re}{p}_k\mathrm{\mid }\ge \mathrm{\mid }{\eta }_k{\mathrm{\mid }}^2-2s\mathrm{\mid }{\eta }_k\mathrm{\mid }\ge \mathrm{\mid }{\eta }_k{\mathrm{\mid }}^2\mathrm{/}2$, so $\mathrm{\mid }{p}_k{\mathrm{\mid }}^{-1}<1\mathrm{/}(8s^2)\le L\mathrm{/}s$ and $\mathrm{\mid }{\eta }_k\mathrm{/}{p}_k\mathrm{\mid }<1\mathrm{/}(2s)$. Thus ${\sup }_k\mathrm{\mid }{p}_k{\mathrm{\mid }}^{-1}\le L\mathrm{/}s$ and ${\sup }_k\mathrm{\mid }{\eta }_k\mathrm{/}{p}_k\mathrm{\mid }\le C_L$. Define $G_{\zeta }f={\sum }_k{p}_k^{-1}(f,w_k)w_k$. Parseval [F1] gives $\mathrm{\parallel }{G}_{\zeta }f{\mathrm{\parallel }}_2\le (L\mathrm{/}s)\mathrm{\parallel }f{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }\mathrm{\nabla }{G}_{\zeta }f{\mathrm{\parallel }}_2\le C_L\mathrm{\parallel }f{\mathrm{\parallel }}_2$. For finite Fourier sums $f_N$, the corresponding partial sums $G_{\zeta }{f}_N$ satisfy $A_{\zeta }{G}_{\zeta }{f}_N=f_N$ termwise. Parseval and the multiplier bounds give $f_N\to f$ in $L^2(Q_L)$ and $G_{\zeta }{f}_N\to G_{\zeta }f$ in $H^1(Q_L)$. Since $A_{\zeta }:H^1(Q_L)\to H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-1}(Q_L)$ continuously, passage to distributions yields $A_{\zeta }{G}_{\zeta }f=f$.

Extend $q$ and $F:=-q$ by zero from $\mathrm{\Omega }$ to $Q_L$. Multiplication by $q$ is bounded on $L^2(Q_L)$, and $\mathrm{\parallel }q{G}_{\zeta }{\mathrm{\parallel }}_{L^2\to L^2}\le \mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}L\mathrm{/}s\mathrm{.}$ Choose ${\tau }_0$ large enough that $s\ge 2L\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $s\ge 1$ whenever $\mathrm{\mid }\zeta \mathrm{\mid }\ge {\tau }_0$. Then the series $\tilde{F}={\sum }_{m=0}^{\mathrm{\infty }}(-q{G}_{\zeta }{)}^{m}F$ converges in $L^2(Q_L)$, satisfies $(I+q{G}_{\zeta })\tilde{F}=F$, and has norm at most $2\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$. Set $r_{\zeta }=G_{\zeta }\tilde{F}$ and restrict it to $\mathrm{\Omega }$. Step 1.1 gives $A_{\zeta }{r}_{\zeta }+q{r}_{\zeta }=F=-q$ on $\mathrm{\Omega }$, as well as $\mathrm{\parallel }{r}_{\zeta }{\mathrm{\parallel }}_2\le (2L\mathrm{/}s)\mathrm{\parallel }q{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }\mathrm{\nabla }{r}_{\zeta }{\mathrm{\parallel }}_2\le 2C_L\mathrm{\parallel }q{\mathrm{\parallel }}_2$.

Since $D=-i\mathrm{\nabla }$, conjugation gives $e^{-i\zeta \cdot x}(-\mathrm{\Delta }+q)e^{i\zeta \cdot x}(1+r_{\zeta })=(D^2+2\zeta \cdot D+\zeta \cdot \zeta +q)(1+r_{\zeta })\mathrm{.}$ The null condition removes $\zeta \cdot \zeta$, the constant amplitude has zero derivatives, and step 2.1 makes the remaining expression $q+A_{\zeta }{r}_{\zeta }+q{r}_{\zeta }=0$. Multiplication by the smooth exponential preserves $H^1(\mathrm{\Omega })$ for this fixed $\zeta$, so $u_{\zeta }\in H^1(\mathrm{\Omega })$ and solves the equation distributionally. The estimates in step 2.1 have $s=\mathrm{\mid }\zeta \mathrm{\mid }\mathrm{/}\sqrt{2}$, giving the asserted bounds after changing the constant.