This is true for your "simple" case. In fact, assuming that the details of my proof below goes through correctly, the following is true.
If g(x,y) ∈ C∞(ℝ2) vanishes on x ≤ 0 then it decomposes as g(x,y) = a(x)G(x,y) where a(x) ∈ C∞(ℝ) vanishes on x ≤ 0 and G(x,y) ∈ C∞(ℝ2).
This can be shown by proving the statements below. I'll state them without full proofs for now, so at least there is a partial argument, and will edit the post to extend the proofs (maybe there are references where these are already proven, but I don't know of any).
First, I'll refer to the following sets of functions.
- U be thet set of functions f(x)∈C∞(ℝ) which vanish on x≤0 and are positive on x>0.
- V be the set of functions f:ℝ+→ℝ such that x-nf(x)→0 as x→0, for each positive integer n.
The statements I need to show the main result are as follows.
Lemma 1: For any f∈V, there is a g∈U such that f(x)/g(x)→0 as x→0.
Sketch proof: Choose any smooth function r:ℝ+→ℝ+ with r(x)≥1 for small x, and r(x)=0 for x≥1. For example, we can use r(x) = exp(2-1/(1-x)) for x≤1. Then, the idea is to choose a sequence of positive reals αk→0 and set
$$ g(x)=x^{\sum_kr_k(x/\alpha_k)}={\rm exp}\left({\rm log}(x)\sum_kr_k(x/\alpha_k)\right)$$ for x>0 and g(x)=0 for x≤0. Only finitely many terms in the summation will be nonzero outside any neighborhood of 0, so it is a well defined expression, and smooth on x>0. The nth derivative is bounded by Knx-ng(x)Σk1{x>αk}αk-n, for positive constants Kn. As long as αk go to zero fast enough, this will be of size O(xn), so g∈C∞(ℝ).
By definition of f∈V, there is a decreasing sequence of positive reals εk such that f(x)≤xn for x≤εn. We just need to make sure that αk≤εn+1 for k≥n to ensure that f(x)/g(x) goes to zero at rate x as x→0.
Lemma 2: For any sequence f1,f2,...∈V there is a g∈U such that fk(x)/g(x)→0 as x→0 for all k.
Lemma 3: For any sequence f1,f2,...∈V there is a g∈U such that fk(x)/g(x)n→0 as x→0 for all positive integers k,n.
The result follows from applying lemma 3 to fi,j,k(x)=max{|(di+j/dxidyj)g(x,y)|:|y|≤k}∈V. details to follow...