Exercise 30, § 3

Let $n\in \mathbf{N}$. Let $(x_1, . . . , x_n)$ denote the dual basis of the canonical basis of $\mathbf{R}^n$.

a) Let $c\geqslant 0$ and $r >0$ be real numbers. There exists a function $u$ defined and analytic in an open neighbourhood U of 0 in $\mathbf{R}^n$ such that

on U and $u(x,0) = 0$ whenever $x\in \mathbf{R}^{n-1}$ satisfies $(x,0)\in U$. (First consider the case $n= 2.$)

b) Let U be an open neighbourhood of 0 in $\mathbf{R}^n$. Let $(a_1, . . . , a_{n-1}, b)$ be functions from $U\times \mathbf{R}$ into $\mathbf{C}$ which are analytic in a neighbourhood of $(0,0)\in U\times \mathbf{R}$. Let Q denote the mapping of the vector space of analytic functions on U into itself such that

There exists at most one function $v$ defined and analytic in an open neighbourhood V of 0 contained in U such that $Q(v) = 0$ and such that $v(x,0) = 0$ for every $x\in \mathbf{R}^{n-1}$ such that $(x,0)\in V$. (Verify that the coefficients of the Taylor expansion at 0 of such a solution $v$ can take only one value.)

c) Let $f$ and $g$ be formal series in $\mathbf{C}[[z_1, . . . , z_n]]$. One says that $f$ majorizes $g$, and one writes $f\ll g$, if

with $|a_{\alpha}|\leqslant b_{\alpha}$ for every $\alpha \in \mathbf{N}^n$. If the formal series $g$ converges in a neighbourhood of 0 in $\mathbf{R}^n$, then the same is true of $f$.

d) If there exist functions $(\widetilde{a}_1, . . . ,\widetilde{a}_{n-1},\widetilde{b})$ analytic in a neighbourhood of 0 such that $a_j\ll \widetilde{a}_j$ for $1\leqslant j\leqslant n-1$ and $b\ll \widetilde{b}$ (in the sense that the corresponding Taylor expansions at 0 satisfy these conditions), and if the equation

in the unknown $v$ has an analytic solution in a neighbourhood of 0, then the equation $Q(v) = 0$ has a unique analytic solution in a neighbourhood of 0. e) There exists a unique solution of the equation $Q(v) = 0$ which is analytic in a neighbourhood of 0 ("Cauchy-Kowalewski theorem").