# Which operators commute with curl?

Let $$X$$ be the space of infinitely differentiable maps from $$\mathbb{R}^3$$ to $$\mathbb{R}^3$$. Let $$C:X\rightarrow X$$ denote the curl map. What are all the linear maps from $$X$$ to $$X$$ that commute with $$C$$? For example, rotations and translations commute with $$C$$. Also of course $$C$$ commutes with itself. Other than products of these three types of linear operators (translation, rotation, and curl) what other operators commute with $$C$$?

1. 'scalar derivative operators' (I don't know a better way to say it), i.e. differential operators that don't see the vector field structure. For example $$\partial_1Cf=C\partial_1f$$. Of course this extends to any $$P(\nabla)$$ where $$\nabla = (\partial_1,\partial_2,\partial_3)$$ and $$P$$ is any scalar valued polynomial of three variables.
2. Convolutions. If $$f$$ is integrable, scalar valued, and compactly supported, then for any $$g\in X$$, the function $$f*g$$ defined by $$f*g(x):=\int_{\operatorname{supp} f} f(y) g(x-y)dy$$ is (1) well-defined (2) an element of $$X$$, and (3) commutes with $$C$$. Point (3) follows because of the standard fact that convolutions commute with any derivative: $$\partial_i (f*g) = f*( \partial_i g)$$ (this fact is of course used to prove (2).)