# In Hamiltonian field theory, do spatial derivatives commute with Poisson brackets?

I'm going back over some of my old notes for a current project, and I'm trying to figure out if I made an error or if I once knew something that I've now forgotten.

Consider a local field theory containing a set of fields $\phi^{(a)}$, for which the Lagrangian density is $$\mathcal{L}(\phi^{(a)}, \dot{\phi}^{(a)}, \partial_i \phi^{(a)} ).$$ Here, $\partial_i$ stands for spatial derivatives only, i.e., we've already done a decomposition into a preferred foliation of spacetime (if necessary.) We can define a set of conjugate field momenta via the usual relation: $$\pi^{(a)} \equiv \frac{\delta \mathcal{L}}{\delta \dot{\phi}^{(a)} }.$$ The Hamiltonian density will then be $$\mathcal{H} = \sum_a \pi^{(a)}\dot{\phi}^{(a)} - \mathcal{L}.$$ We can then define a Poisson bracket for field quantities, of the form $$\{ f, g \} \equiv \sum_a \left[ \frac{ \delta f}{\delta \phi^{(a)}} \frac{ \delta g}{\delta \pi^{(a)}} - \frac{ \delta g}{\delta \phi^{(a)}} \frac{ \delta f}{\delta \pi^{(a)}}\right],$$ where $f$ and $g$ are in principle any two quantities that depend on the fields and the momenta.

Here are my questions:

1. Is it always the case under these definitions that $$\{ \partial_i f, g \} \overset{?}{=} \partial_i \{f, g\}$$ for any two quantities $f$ and $g$?
2. Is it always the case under these definitions that $$\{ \partial_i f, \mathcal{H} \} \overset{?}{=} \partial_i \{f, \mathcal{H} \}$$ for any quantity $f$?

1. Due to OP's mentioned context, we interpret $f$ & $g$ as functions (as opposed to functionals). In particular OP seems to be considering 'same-spacetime' functional derivative (FD), cf. my Phys.SE answer here.
2. Leibniz rule $$\{d_x f(x), g(x)\} +\{f(x), d_xg(x)\}~\stackrel{?}{=}~d_x \{f(x),g(x)\}$$ is not always satisfied.
3. Counterexample: In a hopefully understandable abuse of notation, let $$f(x)~:=~f(\phi(x))\qquad \text{and} \qquad g(x)~:=~g(\pi(x)).$$ Then $$\{d_x f(x), g(x)\} ~=~ 0~=~\{f(x), d_xg(x)\}$$ while $$d_x\{f(x),g(x)\} ~=~ d_x\left(f^{\prime}(\phi(x)) g^{\prime}(\pi(x))\right).$$