Proposition 14.4.12AD3CD874EA4

Proof. Fix an integer $k\geq 0$ and a real exponent $0<\alpha <1$. Consider the map

on the open set of potentials in $C^{k+2,\alpha }$ with $\omega _ u>0$, restricted to $\int _ Xu\, \mathrm{d}V=0$. Its derivative in $(u,b)$ is

Given $h\in C^{k,\alpha }$, integrating $L_ uv-a=h$ against $\mu _ u$ forces $a=-\int _ Xh\, \mathrm{d}\mu _ u$. For this value, the equation for $v$ satisfies the compatibility condition of proposition 11.23.2, and it has a unique solution with zero $\mathrm{d}V$-mean. Schauder estimates show that this inverse is bounded. The implicit function theorem in Banach spaces therefore gives a smooth local solution map around any given solution; this step does not use the global existence theorem. Uniqueness identifies the local maps wherever they overlap, and elliptic regularity allows arbitrary $k$ for smooth data. $\square$