The Laplace fundamental solution is harmonic off its pole

For any real $\alpha$, induction on $j$ using [F12] and [F7] gives $(t^{\alpha }{)}^{(j)}=c_j{t}^{\alpha -j}$ on $(0,\mathrm{\infty })$, where $c_0=1$ and $c_{j+1}=c_j(\alpha -j)$. Each derivative is continuous by the real-power continuity in [F7], so $t^{\alpha }$ is smooth under [F2]. Also ${\log }^{\mathrm{\prime }}(t)=t^{-1}$ by [F8]; applying the same derivative calculation to $t^{-1}$ shows every higher derivative of $\log t$ exists and is continuous. Thus both radial profiles used in [F1] are smooth for $t>0$.

On $U={\mathbb{R}}^n\setminus \{0\}$, put $s(x)={\sum }_{i=1}^n{x}_i^2$. Its coordinate functions and their finite sums and products are smooth by direct coordinate differentiation and [F2]–[F4]. Since $s(x)>0$ on $U$, the radius $r(x)=s(x{)}^{1\mathrm{/}2}$ is smooth there by [F7], step 1.1, and closure under composition [F4]. Composing $r$ with the power profile for $n\ge 3$ or the logarithm profile for $n=2$ proves that $\mathrm{\Phi }$ is smooth on $U$.

For a smooth radial profile $q(r)$ and $r=\mathrm{\mid }x\mathrm{\mid }>0$, the chain rule [F5] and partial-derivative formula [F6] give ${\mathrm{\partial }}_{i}q(r)=q^{\mathrm{\prime }}(r)x_i\mathrm{/}r$. Differentiating again by the chain and product rules [F5], [F7] and [F9] gives ${\mathrm{\partial }}_i^{2}q(r)=q^{\mathrm{\prime }\mathrm{\prime }}(r)x_i^2\mathrm{/}{r}^2+q^{\mathrm{\prime }}(r)(1\mathrm{/}r-x_i^2\mathrm{/}{r}^3)$. Summing over $i$ and using ${\sum }_i{x}_i^2=r^2$ and the Laplacian definition [F10] yields $\mathrm{\Delta }q(r)=q^{\mathrm{\prime }\mathrm{\prime }}(r)+(n-1)q^{\mathrm{\prime }}(r)\mathrm{/}r$. The calculation is on $r>0$, where all derivatives used exist by steps 1.1 and 2.1.

If $n\ge 3$, set $q(r)=r^{2-n}\mathrm{/}((n-2){\omega }_{n-1})$. Then $q^{\mathrm{\prime }}(r)=-r^{1-n}\mathrm{/}{\omega }_{n-1}$ and $q^{\mathrm{\prime }\mathrm{\prime }}(r)=(n-1)r^{-n}\mathrm{/}{\omega }_{n-1}$ by [F7] and [F9], so step 3.1 gives $\mathrm{\Delta }\mathrm{\Phi }=0$. If $n=2$, set $q(r)=-(2\pi {)}^{-1}\log r$. Then $q^{\mathrm{\prime }}(r)=-(2\pi r{)}^{-1}$ by [F8]–[F9] and $q^{\mathrm{\prime }\mathrm{\prime }}(r)=(2\pi r^2{)}^{-1}$ by applying [F7] to $r^{-1}$; hence $q^{\mathrm{\prime }\mathrm{\prime }}+q^{\mathrm{\prime }}\mathrm{/}r=0$. These cases exhaust $n\ge 2$.

For fixed $y$, translation $x\mapsto x-y$ has affine coordinate functions, so direct differentiation gives its identity derivative and zero higher derivatives. The chain rule [F5] therefore gives ${\mathrm{\Delta }}_x(\mathrm{\Phi }(x-y))=(\mathrm{\Delta }\mathrm{\Phi })(x-y)=0$ whenever $x\mathrm{\ne }y$. The translated function is smooth there by [F4], hence is harmonic by [F11]. The assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ in [A1] is carried from the kernel convention but is not used in these pointwise derivative calculations.