Exterior spherical multipole expansion and the ball boundary series

Zonal expansion with uniform tail. Fix $x=rn$ with $r>a$ and let $N\ge 0$. By [L1] the kernel expansion $\mathrm{\mid }x-y{\mathrm{\mid }}^{-1}={\sum }_{l\ge 0}\mathrm{\mid }y{\mathrm{\mid }}^l{r}^{-l-1}{P}_l(n\cdot y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })$ converges absolutely and uniformly for $y\in {\bar{B}}_a(0)$ (the term at $y=0$ being read by the stated homogeneous extension), and its remainder after degree $N$ is bounded by $\mathrm{\mid }y{\mathrm{\mid }}^{N+1}{r}^{-N-2}(1-\mathrm{\mid }y\mathrm{\mid }\mathrm{/}r{)}^{-1}$; multiplying by the $L^1$ density $\rho$ and integrating, the uniform bound gives $\mid \int \rho (y)\mathrm{/}\mathrm{\mid }x-y\mathrm{\mid }dy-{\sum }_{l\le N}{r}^{-l-1}\int \mathrm{\mid }y{\mathrm{\mid }}^l{P}_l(n\cdot y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })\rho (y)dy\mid \le \mathrm{\parallel }\rho {\mathrm{\parallel }}_1{a}^{N+1}{r}^{-N-2}(1-a\mathrm{/}r{)}^{-1}$, so $F(x):=\int \rho (y)\mathrm{/}\mathrm{\mid }x-y\mathrm{\mid }dy={\sum }_{l=0}^{\mathrm{\infty }}{r}^{-l-1}\int \mathrm{\mid }y{\mathrm{\mid }}^l{P}_l(n\cdot y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })\rho (y)dy$, the zonal series converging absolutely and, for every $r_0>a$, uniformly on $\{\mathrm{\mid }x\mathrm{\mid }\ge r_0\}$ with the stated tail bound $\mathrm{\parallel }\rho {\mathrm{\parallel }}_1{a}^{N+1}\mathrm{/}(r_0^{N+2}(1-a\mathrm{/}{r}_0))$.

Ball boundary branch. Part (c) of the Statement is exactly [L3]: the series $u(rn)={\sum }_{l,m}{r}^l{f}_{lm}{Y}_{lm}(n)$ and all its derivatives converge uniformly on each closed smaller ball, $u$ is harmonic, $u(r\cdot )\to f$ in $L^2(S^2)$, uniqueness holds in the class with finite radial $L^2$ supremum, and for continuous $f$ the Poisson representation and the continuous extension to the closed ball hold.

Spherical form, absolute convergence and the degree tail. For $y\mathrm{\ne }0$ the addition theorem [L2] gives $P_l(n\cdot y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })=\frac{4\pi }{2l+1}{\sum }_{m=1}^{2l+1}{Y}_{lm}(n)Y_{lm}(y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })$. For each $l$ put $q_{lm}:=\int \mathrm{\mid }y{\mathrm{\mid }}^l{Y}_{lm}(y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })\rho (y)dy$, a finite integral because $\mathrm{\mid }{Y}_{lm}\mathrm{\mid }$ is bounded on $S^2$ and $\rho \in L^1$; then by [L2] and [L5], and by the monotonicity rule [L6] applied to the absolute values, ${\sum }_m\mathrm{\mid }{Y}_{lm}(n)q_{lm}\mathrm{\mid }\le \int \mathrm{\mid }y{\mathrm{\mid }}^l\mathrm{\mid }\rho (y)\mathrm{\mid }{\sum }_m\mathrm{\mid }{Y}_{lm}(n)Y_{lm}(y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })\mathrm{\mid }dy\le \int \mathrm{\mid }y{\mathrm{\mid }}^l\mathrm{\mid }\rho (y)\mathrm{\mid }\frac{2l+1}{4\pi }dy\le \frac{2l+1}{4\pi }{a}^l\mathrm{\parallel }\rho {\mathrm{\parallel }}_1,$ so the degree-$l$ contribution of the spherical series is bounded in modulus by $(a\mathrm{/}r{)}^l\mathrm{\parallel }\rho {\mathrm{\parallel }}_1\mathrm{/}r$ and the series ${\sum }_l{\sum }_m\frac{4\pi }{2l+1}{Y}_{lm}(n)r^{-l-1}{q}_{lm}$ converges absolutely, with degree tail after $N$ bounded by $\mathrm{\parallel }\rho {\mathrm{\parallel }}_1{r}_0^{-1}{\sum }_{l>N}(a\mathrm{/}{r}_0{)}^l=\mathrm{\parallel }\rho {\mathrm{\parallel }}_1{a}^{N+1}\mathrm{/}(r_0^{N+2}(1-a\mathrm{/}{r}_0))$. Substituting the addition theorem degree by degree into step 1.1 gives, for every $N$, the equality of the truncated zonal and spherical sums, and both truncations converge to the same limit by step 1.1 and this bound; this proves the expansion of part (a) and its uniform convergence on $\{\mathrm{\mid }x\mathrm{\mid }\ge r_0\}$.

The degree-$l$ term is the angular projection. Fix $r>a$ and put $F_r(n):=\int \mathrm{\mid }rn-y{\mathrm{\mid }}^{-1}\rho (y)dy$. Then $\mathrm{\mid }{F}_r(n)\mathrm{\mid }\le \mathrm{\parallel }\rho {\mathrm{\parallel }}_1\mathrm{/}(r-a)$ for every $n$, and $F_r$ is continuous: for $n,n^{\mathrm{\prime }}\in S^2$ and $\mathrm{\mid }y\mathrm{\mid }\le a$, the reverse triangle inequality gives $\mid \mathrm{\mid }rn-y\mathrm{\mid }-\mathrm{\mid }r{n}^{\mathrm{\prime }}-y\mathrm{\mid }\mid \le r\mathrm{\mid }n-n^{\mathrm{\prime }}\mathrm{\mid }$, while both denominators are at least $r-a$; hence $\mathrm{\mid }{F}_r(n)-F_r(n^{\mathrm{\prime }})\mathrm{\mid }\le r\mathrm{\parallel }\rho {\mathrm{\parallel }}_1\mathrm{\mid }n-n^{\mathrm{\prime }}\mathrm{\mid }\mathrm{/}(r-a{)}^2$, proving continuity directly. Hence $F_r\in L^2(S^2,d\sigma )$ with $\mathrm{\parallel }{F}_r{\mathrm{\parallel }}_2^2\le 4\pi \mathrm{\parallel }\rho {\mathrm{\parallel }}_1^2\mathrm{/}(r-a{)}^2$. By step 2.1 the spherical series for $F_r$ converges uniformly on the compact sphere $S^2$, so the pairing with each basis element may be taken termwise and $⟨F_r,Y_{km}⟩={\sum }_{l,m^{\mathrm{\prime }}}\frac{4\pi }{2l+1}{r}^{-l-1}{q}_{l{m}^{\mathrm{\prime }}}⟨Y_{l{m}^{\mathrm{\prime }}},Y_{km}⟩=\frac{4\pi }{2k+1}{r}^{-k-1}{q}_{km}$ by the orthonormality of the basis [L7]; this is precisely the stated $L^2(S^2,d\sigma )$ angular projection.

Coulomb and dipole identification. Summing the degree-$l$ spherical term over $m$ and using the addition theorem in reverse gives $r^{-l-1}{\sum }_m\frac{4\pi }{2l+1}{Y}_{lm}(n)q_{lm}=r^{-l-1}\int \mathrm{\mid }y{\mathrm{\mid }}^l{P}_l(n\cdot y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })\rho (y)dy$, the zonal degree term computed in step 1.1, and it is the degree-$l$ term of the spherical series of step 2.1. Since $P_0=1$ and $P_1(z)=z$ [L1], the degrees $l=0$ and $l=1$ contribute $r^{-1}\int \rho =Q\mathrm{/}r$ and $r^{-2}\int (n\cdot y)\rho (y)dy=p\cdot x\mathrm{/}{r}^3$; the remaining tail is $F(x)-Q\mathrm{/}r-p\cdot x\mathrm{/}{r}^3$, and for $r\ge 2a$ the explicit remainder bound $\mathrm{\mid }R(x)\mathrm{\mid }\le 16a^2\mathrm{\parallel }\rho {\mathrm{\parallel }}_1\mathrm{/}{r}^3$ of [L4] applies to it, so the degree-zero and degree-one terms are the Coulomb and dipole terms of the Statement.

Termwise differentiation. Each degree term is $U_l(x):=r^{-l-1}\int \mathrm{\mid }y{\mathrm{\mid }}^l{P}_l(n\cdot y\mathrm{/}\mathrm{\mid }y\mathrm{\mid })\rho (y)dy=\int \rho (y)c_l(x,y)dy$, with the full weighted coefficient $c_l$ of [L1]. For fixed $l$, its polynomial extension in $y$ is smooth jointly for $x\mathrm{\ne }0$, including $y=0$, and harmonic in $x$ by [L1]. On a compact observation neighbourhood times ${\bar{B}}_a(0)$ its spatial derivatives are uniformly continuous and bounded. By [L9] along coordinate segments, for each such derivative $D^{\beta }{c}_l$ its difference quotient converges uniformly in $y$ to ${\mathrm{\partial }}_i{D}^{\beta }{c}_l$; integrating the error bounds it by $\mathrm{\parallel }\rho {\mathrm{\parallel }}_1$ times that uniform error. Induction therefore gives $D^{\beta }{U}_l=\int \rho D^{\beta }{c}_l$, continuous by the same uniform bounds, and $\mathrm{\Delta }{U}_l=\int \rho {\mathrm{\Delta }}_x{c}_l=0$. For a nonempty compact $K$ in $\{\mathrm{\mid }x\mathrm{\mid }>a\}$ choose nested compact neighbourhoods $K\subset \mathrm{int}{K}_1\subset K_1\subset \mathrm{int}{K}_2\subset \{\mathrm{\mid }x\mathrm{\mid }>a\}$ and a fixed $\delta >0$ such that ${\bar{B}}_{\delta }(x)\subset K_2$ for every $x\in K_1$. The harmonic partial sums $S_N={\sum }_{l\le N}{U}_l$ converge uniformly on $K_2$ by step 2.1. Hence [L8] applied to $S_M-S_N$ gives ${\sup }_{K_1}\mathrm{\mid }{D}^{\alpha }(S_M-S_N)\mathrm{\mid }\le C_{\alpha }{\delta }^{-3-\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{\mid }{B}_{\delta }\mathrm{\mid }{\sup }_{K_2}\mathrm{\mid }{S}_M-S_N\mathrm{\mid }\to 0$ for every multi-index $\alpha$. The derivative limits are uniform and continuous on $K_1$. Apply [L9] to $D^{\beta }{S}_N$ on short coordinate segments in $\mathrm{int}{K}_1$ and pass to the uniform limits in the integral identity; this identifies the limit of ${\mathrm{\partial }}_i{D}^{\beta }{S}_N$ with the corresponding derivative of the limit of $D^{\beta }{S}_N$. Induction proves termwise differentiation to every order on $K$, with $F$ smooth in the sense of [L10]. The empty compact case is vacuous.

Assembly. Part (a) is steps 1.1, 2.1, 3.1 and 3.2 (expansion, uniform tail, angular projection and Coulomb–dipole identification); part (b) is step 3.3; part (c) is step 1.2; the statement about the value at $y=0$ is the finite-assignment convention recorded in the Statement, which leaves every integral unchanged because a singleton is null. This proves all three parts.