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 $C$?


I don't claim to produce an exhaustive list, but some notable ones that are missing:

  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).)

If you include enough decay at infinity, you also get lots of others like (scalar) Fourier multipliers and the projection to the gradient or curl parts of the Helmholtz decomposition.


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