When working with the commutator of two bounded operators, there is a simple if slightly crude bound on the norm of the commutator:
$$ \lVert[A, B]\rVert = \lVert AB - BA \rVert \leq 2\lVert A \rVert \lVert B \rVert $$.
However, I am currently working with unbounded functions and their two-norms in the context of classical mechanics, and was wondering if there was a bound for the Poisson bracket similar in spirit; something of the form
$$ \lVert \{f, g \} \rVert_2 = \lVert \nabla_\omega f \nabla_q g - \nabla_q f \nabla_\omega g \rVert_2 \leq 2 \lVert f \rVert_2 \lVert g \rVert_2$$.
I have tried looking for this by using the definition of the two-norm, but do not seem to get any useful results apart from rewriting the definitions.