Lemma from Donnelly-Fefferman's paper

Question

I am reading the paper Nodal sets of eigenfunctions of the Laplacian on Surfaces by Donnelly and Fefferman available here. I have a problem understanding Lemma 5.10. To my understanding, what follows is the content of the Lemma 5.10

Let $\mathbb{D}$ be the unit disc in $\mathbb{R}^2$ centered at the origin. Fix $m \in \mathbb{N}$. Then there is a constant $C_m >0$ such that for any smooth $g :\mathbb{D} \rightarrow \mathbb{R}$ satisfying the following condition \begin{align*} \max_{|\alpha| \leq m} \left| \frac{\partial^{\alpha}g}{\partial z^{\alpha}}(0) \right| \geq 1 \end{align*} there are a pair of perpendicular directions $v_1$ and $v_2$ and some $k \leq m$ (here $v_1, v_2$ and $k$ can possibly depend on the function $g$) such that the following holds \begin{align*} |\partial_{v_1}^{k}g(0)| \geq C_m~\mbox{and}~|\partial_{v_2}^{k}g(0)| \geq C_m \end{align*}

Here are my questions regarding this Lemma.

1. Firstly, is my understanding of the statement of Lemma 5.10 correct? More specifically, is it claimed that there is a universal constant $C_m$ (independent of the function $g$)?

2. If the answer to my first question is Yes, I cannot really see how the proof provided in the paper establishes existence of the universal constant $C_m$.

Thanks in advance!

Answer 1

$\newcommand{\al}{\alpha} \newcommand{\G}{\mathcal{G}} \newcommand{\PP}{\mathcal{P}}$ You are missing condition (ii) in formula (5.9) in that paper. That formula is \begin{equation} \begin{aligned} &\text{(i)}\quad \sup_{|\al|\le c_4} \Big| \frac{\partial^{\al}g}{\partial z^{\al}}(0) \Big|\ge b_6, \\ &\text{(ii)}\quad \sup_{|z|<1/2}\sup_{|\al|\le c_4+1} \Big| \frac{\partial^{\al}g}{\partial z^{\al}}(z)\Big|\le b_7. \end{aligned} \tag{5.9} \end{equation} It appears that, in the paper, the $a_i$'s, $b_i$'s, and $c_i$'s denote positive real constants, which may depend only on such constants previously mentioned in the paper.

Then Lemma 5.10 there seems to say that for some nonnegative integer $k\le c_4$ and some perpendicular unit vectors $v_1$ and $v_2$ we have \begin{align*} |\partial_{v_1}^{k}g(0)|\ge b_{11}~\mbox{and}~|\partial_{v_2}^{k}g(0)|\ge b_{11}. \end{align*}

This is indeed true, with $b_{11}>0$ depending only on $c_4,b_6,b_7$. Indeed, let $\G$ be the set of all smooth functions $g$ satisfying (5.9), and let $\PP$ be the set of the Taylor polynomials (at $0$) of order $\lfloor c_4\rfloor$ of the functions $g\in\G$. The set $\PP$ is obviously compact in the natural topology on $\PP$. So, there exists \begin{align} \mu&:=\min_{g\in\G}\max_{k,v_1,v_2}(|\partial_{v_1}^{k}g(0)|\wedge|\partial_{v_2}^{k}g(0)|) \\ & =\min_{P\in\PP}\max_{k,v_1,v_2}(|\partial_{v_1}^{k}P(0)|\wedge|\partial_{v_2}^{k}P(0)|)\in[0,\infty), \end{align} where $\max_{k,v_1,v_2}$ is over all nonnegative integers $k\le c_4$, and all perpendicular unit vectors $v_1$ and $v_2$.

Let $P$ be a minimizer for $\min_{P\in\PP}$. By condition (5.9)(i), $P$ is a nonzero polynomial of degree $\le c_4$ and hence has a a nonvanishing term of order $k\le c_4$. So, $\partial_v^{k}P(0)=0$ only for finitely many unit vectors $v$. So, we can find perpendicular unit vectors $v_1$ and $v_2$ such that $|\partial_{v_1}^{k}P(0)|\wedge|\partial_{v_2}^{k}P(0)|>0$. Thus, $\mu>0$. It remains to let $b_{11}:=\mu>0$, which depends only on $c_4,b_6,b_7$.