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It is well-known that if $p$ is a homogeneous polynomial of degree $k$ in $n$ variables, $p∈\mathcal{P}_k$, then the ordinary derivative $p(∇) (r^{2−n})$ has the form $A_{n,k}\Upsilon(x)r^{2−n−2k}$ where $A_{n,k}$ is a constant and where $\Upsilon$ is a harmonic homogeneous polynomial of degree $k,$ $\Upsilon\in \mathcal{H}_k$, actually the projection of $p$ onto $\mathcal{H}_k$. Here we study the distributional
derivative $p(\overline{∇})(r^{2−n})$ and show that the ordinary part is still a multiple of $\Upsilon$, but that
the delta part is independent of $\Upsilon$, that is, it depends only on $p−\Upsilon$. We also show that
the exponent $2−n$ is special in the sense that the corresponding results for $p(∇)(r^α)$ do not hold if $α\neq 2−n$.

Furthermore, we establish that harmonic polynomials appear as multiples of $r^{2−n−2k−2k'}$ when $p(∇)$ is applied to harmonic multipoles of the form $\Upsilon'(x)r^{2−n−2k'}$ for some $\Upsilon' \in \mathcal{H}_k$.

It is well-known that if $p$ is a homogeneous polynomial of degree $k$ in $n$ variables, $p∈\mathcal{P}_k$, then the ordinary derivative $p(∇) (r^{2−n})$ has the form $A_{n,k}\Upsilon(x)r^{2−n−2k}$ where $A_{n,k}$ is a constant and where $\Upsilon$ is a harmonic homogeneous polynomial of degree $k,$ $\Upsilon\in \mathcal{H}_k$, actually the projection of $p$ onto $\mathcal{H}_k$. Here we study the distributional
derivative $p(\overline{∇})(r^{2−n})$ and show that the ordinary part is still a multiple of $\Upsilon$, but that
the delta part is independent of $\Upsilon$, that is, it depends only on $p−\Upsilon$. We also show that
the exponent $2−n$ is special in the sense that the corresponding results for $p(∇)(r^α)$ do not hold if $α\neq 2−n$.

Furthermore, we establish that harmonic polynomials appear as multiples of $r^{2−n−2k−2k'}$ when $p(∇)$ is applied to harmonic multipoles of the form $\Upsilon'(x)r^{2−n−2k'}$ for some $\Upsilon' \in \mathcal{H}_k$.

*Analysis in Theory and Applications*.

*34*(4). 336-347. doi:10.4208/ata.OA-2017-0062