I've been reading Range's Holomorphic Functions and Integral Representations in Several Complex Variables and I am currently stuck on some techincal details in the proof of Lemma 2.19:
A bounded Levi pseudoconvex domain $D$ with $C^3$ boundary is pseudoconvex (i.e. has a strictly plurisubharmonic exhaustion function).
But my issues boil down to the following: Suppose we have a $C^3$ defining function $r$ defined on some open set $U$ containing a point $P\in bD$. Define the Levi form as (with $t\in\mathbb{C}^n$) $$ L_z(r,t)=\sum_{j,k=1}^{n}\frac{\partial^2r}{\partial z_j\partial \bar{z}_k}(z)t_j\bar{t}_k. $$ Fix $w\in\mathbb{C}^n$. At each $z\in bD$, we may orthogonally decompose $w$ as $w=w'_z+w''_z$, where $w_z'\in T^{\mathbb{C}}_z(bD)$ and $w_z''\in T_z^{\mathbb{C}}(bD)^{\perp}$, and furthermore $w'_z$ and $w''_z$ both depend differentiably on $z$. I want to show that, by shrinking $U$ if necessary, $$ L_z(r,w_z')\geq -C|r(z)||w_z'|^2. $$ for some positive constant $C>0$. Supposedly this is possible since $z\mapsto L_z(r,w_z')$ belongs to $C^1$. I've thought about this for an embarrassingly long time. Here are my ideas so far.
Ideas
Consider the function $f:U\to\mathbb{R}$ given by $$ f(z)=L_z(r,w_z') $$ Fix $\zeta\in bD$. By Taylor's theorem, we have $$ f(z)=f(\zeta)+O(|z-\zeta|)\geq O(|z-\zeta|), $$ where the inequality follows from pseudoconvexity. This implies there exists a $C>0$ such that $$ f(z)=L_z(r,w'_z)=\sum_{j,k=1}^{n}\frac{\partial^2r}{\partial z_j\partial\bar{z}_k}(w'_z)_j\overline{(w'_z)_k}\geq -C|z-\zeta|. $$ Now from here I'm tempted to use the fact that $|z-\zeta|\geq\text{dist}(z,bD)=-h(z)|r(z)|$ for some positive function $h$, but with the negative sign in front of the $C$ this inequality is useless.
I'm sure I'm overthinking things and there's probably an easier way. Any help is greatly appreciated. Thank you.
$w'_z$ is the projection of $w$ onto the tangent space of $r$ at $z.$ Since we're just talking about a bilinear form, it suffices to show $L_z(r,w'_z)\geq -C|r(z)|$ for $|w'_z|\approx 1$ (e.g. $\tfrac12\leq|w'_z|\leq 2$). Shrink $U$ to a bounded set. So we can consider $r$ and $w'_z$ to be bounded which makes things easier. $L_z(r,w'_z)$ is a $C^1$ function in $z$ on a bounded set, so it's bounded below by some negative constant multiplied by the distance to any point in $bD.$ This means it's at least $-C|r(z)|\max(|\partial r/\partial z|^{-1})$.