4 slightly restructured, change subject as suggested by mathphysicist

# Differential operators preserving the space of harmonic functions (aka higher symmetries oftheLaplacian)

The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential operator $\delta$ such that $\Delta D = \delta \Delta$. Obviously these operators preserve the kernel of $\Delta$, i.e. the space of harmonic functions. The mentioned article finds essentially all such operators $D$. The result is that up to trivial operators $D = P\Delta$ all the operators $D$ have polynomial coefficients and are generated by sums of compositions of first order operators of this kind.

Now let

First question: Let $D$ be any differential operator preserving the space of harmonic functions.

First question: It is easy to see that the operator $\delta = \Delta D (\Delta)^{-1}$ is well defined and satisfies $\Delta D = \delta \Delta$. Is $\delta$ also a differential operator?

Second question: Is it true that all differential operators, which preserve the space of harmonic functions, are generated by first order ones with this property?

One can also ask these questions only for linear differential operators or for operators from the Weyl algebra (i.e. linear differential operators with polynomial coefficients). For example, by a theorem of Peetre, the answer to the first question is affirmative if the operator $\delta = \Delta D (\Delta)^{-1}$ is local (i.e. the support of $\delta u$ is contained in the support of $u$).

Third question: What makes the linked article so interesting that it was published in Annals?

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