Weighted Hölder Poisson inversion and openness of the normalized volume equation

We first specify the function-space facts needed below. Choose finitely many coordinate balls with larger concentric balls still in the charts and compact inner balls covering $X$. Changes of coordinates and their derivatives are bounded on these finitely many compact overlaps, so the chain rule and $[ab{]}_{\alpha }\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}[b{]}_{\alpha }+\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }}[a{]}_{\alpha }$ compare any two such norms. If a sequence is Cauchy in $C^{r,\alpha }$, every derivative through order $r$ converges uniformly on each compact ball. Integration on coordinate segments shows that these limits are the successive derivatives of the limiting function. The top-order difference quotients obey the Cauchy bounds on passage to the limit, proving convergence in the full norm. Overlap identities pass uniformly, so these limits define one function on $X$. Thus the global spaces are Banach. Integration against a fixed smooth density is continuous, so the indicated mean-zero subspaces are closed and Banach. Repeated Leibniz and the displayed product estimate make multiplication a continuous bilinear map on each $C^{r,\alpha }$. Smooth coefficients act boundedly.

The real principal matrix of ${\mathrm{\Delta }}_1$ is positive definite on every compact chart, and its coefficients are smooth. Applying [F3] on finitely many inner/outer balls gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{2,\alpha }}\le C(\mathrm{\parallel }{\mathrm{\Delta }}_{1}u{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }})$ for smooth $u$. Higher orders follow without assuming a new higher-order estimate: for $L_0=a^{ij}{\mathrm{\partial }}_{ij}+b^i{\mathrm{\partial }}_i$ and a coordinate derivative $D^{\beta }$, the equation for $D^{\beta }u$ is $L_0{D}^{\beta }u=D^{\beta }(L_{0}u)-{\sum }_{0<\gamma \le \beta }\left(\genfrac{}{}{0px}{}{\beta }{\gamma }\right)(D^{\gamma }{a}^{ij})D^{\beta -\gamma }{\mathrm{\partial }}_{ij}u-{\sum }_{0<\gamma \le \beta }\left(\genfrac{}{}{0px}{}{\beta }{\gamma }\right)(D^{\gamma }{b}^i)D^{\beta -\gamma }{\mathrm{\partial }}_{i}u$. Every commutator term contains at most $\mathrm{\mid }\beta \mathrm{\mid }+1$ derivatives of $u$. Inductively use the already obtained $C^{\mathrm{\mid }\beta \mathrm{\mid }+1,\alpha }$ estimate on intermediate balls, and [F3] on smaller balls, to obtain the claimed $C^{k+2,\alpha }$ estimate, with a constant depending on finitely many coefficient norms and on the chosen fixed nested radii. For every fixed $k$ these balls can be chosen in the original atlas and their innermost subballs still cover $X$. This proves the unweighted global estimate in the Statement.

Stokes gives $-{\int }_{X}u{\mathrm{\Delta }}_{1}ud{V}_1={\int }_X\mathrm{\mid }\mathrm{\partial }u{\mathrm{\mid }}_{{\omega }_1}^{2}d{V}_1$, so the kernel on smooth real functions consists exactly of the constants, by connectedness. The positive-frequency symbol of $P=-{\mathrm{\Delta }}_1$ is scalar positive definite; $P$ is formally symmetric for $d{V}_1$. Hence [F2] applies with real trivial line bundle and precisely its $H^2$ domain, in real dimension $2n\ge 2$. For a smooth $f$ of zero $d{V}_0$-mean, put $h=f\mathrm{/}\rho$. Its compatibility condition is exactly ${\int }_{X}hd{V}_1={\int }_{X}fd{V}_0=0$. Let $v=G(-h)$, so ${\mathrm{\Delta }}_{1}v=h$ and $v$ is smooth of zero $d{V}_1$-mean. Set $u=v-({\int }_{X}vd{V}_0)\mathrm{/}({\int }_{X}d{V}_0)$. Then $Lu=f$ and $u$ has the required reference mean zero. The $L^2\to H^2$ Green bound, equivalence of the two smooth positive densities, and Cauchy–Schwarz for this constant correction give $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(d{V}_0)}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(d{V}_0)}$. Thus no wrong-volume orthogonality condition has been imposed.

To remove the $C^0$ term from the estimate, a uniform coordinate-ball averaging argument gives, for every $\epsilon >0$, $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}\le \epsilon \mathrm{\parallel }u{\mathrm{\parallel }}_{C^{k+2,\alpha }}+C_{\epsilon }\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(d{V}_0)}$. Indeed at a maximum-modulus point choose one of the inner charts. On a surrounding coordinate ball of radius $r$ lying in its outer chart, the mean value theorem bounds the difference between the central value and the ball average by $Cr\mathrm{\parallel }Du{\mathrm{\parallel }}_{\mathrm{\infty }}$, while Cauchy–Schwarz bounds the average by $C{r}^{-n}\mathrm{\parallel }u{\mathrm{\parallel }}_2$; coordinate densities are uniformly comparable to Lebesgue measure. Choose $r$ uniformly small so the first coefficient is at most $\epsilon$. Insert this inequality into step 2.1, using bounded multiplication by ${\rho }^{-1}$ and then the $L^2$ bound from step 2.2. Absorbing with $\epsilon$ smaller than the reciprocal of twice the Schauder constant gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{k+2,\alpha }}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{k,\alpha }}$ for the smooth normalized solution. All constants here concern the fixed smooth metrics, not a varying degenerating family.

Now let $f\in C_0^{k,\alpha }$ without a smoothness assumption. Subordinate cutoffs and convolution on the outer charts produce smooth $f_j$ with uniformly bounded $C^{k,\alpha }$ norms and $f_j\to f$ in $C^{k,\beta }$ for every $0<\beta <\alpha$; subtract their $d{V}_0$ averages. The bounds follow by differentiating convolution and the identity $\mathrm{\mid }{\eta }_{ϵ}\ast w(x)-{\eta }_{ϵ}\ast w(y)\mathrm{\mid }\le [w{]}_{\alpha }\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$ for a nonnegative unit-mass mollifier. Uniform convergence of each derivative through order $k$ follows from continuity on compact supports; the interpolation split into $\mathrm{\mid }x-y\mathrm{\mid }\le \delta$ and $\mathrm{\mid }x-y\mathrm{\mid }>\delta$ then gives convergence at every smaller exponent. We do not assert smooth density in the full $C^{k,\alpha }$ norm. Solve for smooth $u_j$ by step 2.2. Step 3.1 bounds them uniformly in $C^{k+2,\alpha }$. The Euclidean Arzelà–Ascoli supplier in [F4], applied to the finitely many derivative arrays and compact chart balls, gives a subsequence converging in $C^{k+2,\beta }$ to a normalized $u$. Each $\alpha$ difference quotient of each top derivative passes to its limit, so $u\in C^{k+2,\alpha }$ with the same bound. The equation passes in $C^0$, giving $Lu=f$. Uniqueness follows from the kernel argument in step 2.2, which also applies to a $C^{k+2,\alpha }$ function by classical integration by parts. Therefore this solution map is linear and bounded, proving the stated Banach isomorphism. This is the full passage from the $L^2$ Fredholm inverse to the Hölder inverse.

The local regularity bootstrap used for openness must also start at the regularity the implicit theorem supplies. Here is its needed form. If $u\in C^{2,\alpha }$ solves a scalar uniformly elliptic equation with coefficients and source in $C^{1,\alpha }$, form the coordinate difference quotient ${\delta }_{h}u(x)=(u(x+h{e}_a)-u(x))\mathrm{/}h$ on a smaller ball. It lies in $C^{2,\alpha }$ for every nonzero $h$. Its equation has source ${\delta }_{h}f-({\delta }_h{a}^{ij})({\mathrm{\partial }}_{ij}u)(x+h{e}_a)-({\delta }_h{b}^i)({\mathrm{\partial }}_{i}u)(x+h{e}_a)-({\delta }_{h}c)u(x+h{e}_a)$ for an operator with the unshifted coefficients. Each coefficient difference quotient is bounded in $C^{0,\alpha }$ by the $C^{1,\alpha }$ norm, using ${\delta }_{h}a={\int }_0^1{\mathrm{\partial }}_{a}a(x+sh{e}_a)ds$; the same holds for $f$. The source norm is uniformly bounded, as is $\mathrm{\parallel }{\delta }_{h}u{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{\mathrm{\partial }}_{a}u{\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus [F3] bounds ${\delta }_{h}u$ uniformly in $C^{2,\alpha }$ on an inner ball. Passing to a subsequence as in step 4.1 shows ${\mathrm{\partial }}_{a}u\in C^{2,\alpha }$. Therefore $u\in C^{3,\alpha }$. Repeating this argument on differentiated equations, with the explicit commutator formula of step 2.1, proves the higher-order regularity and estimates whenever the corresponding coefficient and source derivatives are available. This avoids applying a classical $C^2$ estimate to a derivative known initially only to be $C^1$.

Define $\mathcal{V}(\varphi )=({\omega }_0+i\mathrm{\partial }\bar{\mathrm{\partial }}\varphi {)}^n\mathrm{/}d{V}_0$ on the open positive cone in $C_0^{3,\alpha }$. It is a polynomial in the coordinate second derivatives of $\varphi$ and fixed smooth coefficients, so step 1.1 makes it a smooth Banach-space map to $C^{1,\alpha }$. It has fixed total integral: the difference of the two top forms is $i\mathrm{\partial }\bar{\mathrm{\partial }}\varphi \wedge {\sum }_{j=0}^{n-1}{\omega }_{\varphi }^j\wedge {\omega }_0^{n-1-j}$, and Stokes makes its integral zero. For $C^{3,\alpha }$ potentials this follows by smooth approximation in $C^2$, preserving positivity, and continuity of the determinants and integrals. At any smooth solution ${\varphi }_{\ast }$, determinant differentiation gives $D{\mathcal{V}}_{{\varphi }_{\ast }}(\psi )={\rho }_{\ast }{\mathrm{\Delta }}_{{\omega }_{{\varphi }_{\ast }}}\psi$, where ${\rho }_{\ast }={\omega }_{{\varphi }_{\ast }}^n\mathrm{/}d{V}_0$. This derivative maps exactly the fixed-reference zero-mean spaces and is a bounded isomorphism by step 4.1 with $k=1$. Apply [F5] with the density difference $y$ as a Banach-space parameter to $\mathcal{V}(\varphi )-{\rho }_{\ast }-y$. It gives normalized positive $C^{3,\alpha }$ solutions for all $y$ sufficiently small in the reference-mean-zero $C^{1,\alpha }$ space. Take $y={\rho }_t-{\rho }_{\ast }$ for sufficiently close path parameters within the given interval; no parameter extension beyond an endpoint is needed. In a holomorphic chart write $h_{a\bar{b}}=g_{a\bar{b}}+{\varphi }_{a\bar{b}}$. Taking a first coordinate derivative of $\log \det h=\log \det g+\log {\rho }_t$ shows that $\psi ={\mathrm{\partial }}_c\varphi$ satisfies $h^{a\bar{b}}{\psi }_{a\bar{b}}={\mathrm{\partial }}_c(\log \det g+\log {\rho }_t)-h^{a\bar{b}}{\mathrm{\partial }}_c{g}_{a\bar{b}}$. The coefficients and source are $C^{1,\alpha }$, while $\psi \in C^{2,\alpha }$; step 5.1 gives $\psi \in C^{3,\alpha }$, hence $\varphi \in C^{4,\alpha }$. Iteration gives smoothness. Thus the smooth solvability set is relatively open.