Positive Fourier inverse for the linearized CMC constraint

For a fixed smooth real direction $v$, its periodic lift is continuous on the nonempty compact box $[0,2\pi {]}^3$ by [F4, F7], so [F8] gives a finite bound $V:=\sup \mathrm{\mid }v\mathrm{\mid }$. If $V>0$, choose $\mathrm{\mid }ϵ\mathrm{\mid }<{\varphi }_0\mathrm{/}(2V)$; if $V=0$, positivity is immediate. In either case ${\varphi }_0+ϵv>0$ for all such directions and sufficiently small $\mathrm{\mid }ϵ\mathrm{\mid }$. Every term in $c$ is positive because $A>0$, ${\varphi }_0>0$ and $\tau \mathrm{\ne }0$. Since ${\tilde{h}}_{ij}=L_{\ast }^2{\delta }_{ij}$, the Christoffel formula in the conformal-data definition gives ${\tilde{\mathrm{\Gamma }}}^k{}_{ij}=0$ in the local periodic charts. The curvature definition then gives $\tilde{R}=0$, and the Laplacian formula gives $\tilde{\mathrm{\Delta }}=L_{\ast }^{-2}{\sum }_i{\mathrm{\partial }}_i^2$. Differentiating the scalar expression [F2] at the constant ${\varphi }_0$ gives $-8\tilde{\mathrm{\Delta }}v$ from its linear Laplacian term, zero from $\tilde{R}=0$, $7A{\varphi }_0^{-8}v$ from $-A{\varphi }^{-7}$, and $(10\mathrm{/}3){\tau }^2{\varphi }_0^{4}v$ from $(2\mathrm{/}3){\tau }^2{\varphi }^5$. Thus its directional linearization is $\mathcal{A}=-8\tilde{\mathrm{\Delta }}+c$. For each $i$, periodicity makes the integral of ${\mathrm{\partial }}_i(v{\mathrm{\partial }}_{i}v)$ over one period zero; expanding that derivative gives $\int v{\mathrm{\partial }}_i^{2}vd\mu =-\int ({\mathrm{\partial }}_{i}v{)}^{2}d\mu$. Summing over $i$ therefore yields $\int v\mathcal{A}vd\mu =8L_{\ast }^{-2}\int \mathrm{\mid }{\mathrm{\nabla }}_{0}v{\mathrm{\mid }}^{2}d\mu +c\int v^{2}d\mu >0$ for every nonzero smooth real $v$: choose a point where $v\mathrm{\ne }0$; continuity gives a coordinate box of positive torus measure on which $\mathrm{\mid }v\mathrm{\mid }$ is bounded below by a positive constant, so $\int v^{2}d\mu >0$. Here ${\mathrm{\nabla }}_{0}v=({\mathrm{\partial }}_{1}v,{\mathrm{\partial }}_{2}v,{\mathrm{\partial }}_{3}v)$.

For any $v\in H^{s+2}$, the weak-derivative Fourier identity [F5] gives $\widehat{\mathcal{A}v}(k)=a_k\widehat{v}(k)$ with $a_k=8L_{\ast }^{-2}\mathrm{\mid }k{\mathrm{\mid }}^2+c$. Since $c>0$, every $a_k>0$, and $a_k\ge m(1+\mathrm{\mid }k{\mathrm{\mid }}^2)$ for $m:=\min \{8L_{\ast }^{-2},c\}>0$; also $a_k\le M(1+\mathrm{\mid }k{\mathrm{\mid }}^2)$ for $M:=\max \{8L_{\ast }^{-2},c\}$. The upper bound proves $N_s(\mathcal{A}v)\le M{N}_{s+2}(v)$. Norm equivalence in [F5] therefore makes the weak operator bounded for the derivative Sobolev norms. For real $f\in H^s$, [F6] gives $\widehat{f}(-k)=\overline{\widehat{f}(k)}$. Since $a_{-k}=a_k$ is real, $\widehat{v}(k):=\widehat{f}(k)\mathrm{/}{a}_k$ has the same conjugate symmetry and therefore [F6] gives it a unique complex $L^2$ representative $v$. The coefficient definition gives $\widehat{\bar{v}}(k)=\overline{\widehat{v}(-k)}=\widehat{v}(k)$, so injectivity of the coefficient isometry [F6] yields $\bar{v}=v$ in $L^2$. The lower bound gives $N_{s+2}(v{)}^2={\sum }_k\frac{(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{s+2}}{a_k^2}\mathrm{\mid }\widehat{f}(k){\mathrm{\mid }}^2\le m^{-2}{N}_s(f{)}^2\mathrm{.}$ The lower bound makes the coefficient sequence square-summable with weight $(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{s+2}$; [F5] then gives $v\in H^{s+2}$, with $\mathcal{A}v=f$. If $\mathcal{A}v=0$, then $a_k\widehat{v}(k)=0$ for every $k$, hence all coefficients vanish and $v=0$. This proves surjectivity, injectivity and the asserted inverse bound for every integer $s\ge 2$.