For a norm function $f$, do we have a bound on $f([X, Y])$ for two vector field $X$ and $Y$?

Question

Since $[X, Y]$ is related to the commutator of the flows generated by $X$ and $Y$, if $X$ and $Y$ are sufficiently smooth, there should be an upper bound on $f([X, Y])$, say a function of $f(X)$, $f(Y)$ and Lipschitz constants of $X$ and $Y$? I couldn't imagine it become arbitrarily large. And is it true that at least one of the inequality $f([X, Y]) < f(X)$, $f([X, Y]) < f(Y)$ holds? If it doesn't hold for general vector field, does it hold in some Lie algebra where the vector fields are left invariant?

Answer 1

Consider $X = \partial_x$ and $Y = \sin\left(e^x\right) \partial_y$ on $\mathbb{R}^2$. Then \begin{align} \|X\| &= 1, & \|Y\| &= |\sin e^x|\leqslant 1, &\|[X,Y]\| = e^x|\cos e^x|. \end{align} It follows that there exists $X$ and $Y$ that have bounded norms but $[X,Y]$ does not, and your intuition is wrong. The fact is that $[X,Y]$ does not depend only on $X$ and $Y$ as vectors (i.e on their pointwise behaviour) but also on their $1$-jet (i.e on their local behaviour). It is well-known that a bounded function can have unbounded derivative: this is analogous.

If moreover $X$ and $Y$ are considered left-invariant, then so is $[X,Y]$ and if the norm is also left-invariant, $\|X\|$, $\|Y\|$ and $\|[X,Y]\|$ are constant functions. If no assumption is made on the norm, I can't see any conclusion.