Strongly continuous rotation action

Given: the sphere $S^2$, the polar measure $\sigma$, the complete orthonormal family $(Y_{lm})$, the blocks $V_l$, the finite span $H$, the spaces $H_l^{\mathrm{p}\mathrm{o}\mathrm{l}}$, the restriction maps ${\mathrm{r}\mathrm{e}\mathrm{s}}_l$, and the group $SO(3)$.

The polar surface measure equals the chart surface measure on $S^2$, is finite with $\sigma (S^2)=4\pi$, and is preserved by orthogonal transformations: $\sigma (OE)=\sigma (E)$ for every orthogonal $O$ and every Borel $E\subseteq S^2$ (Agreement with the existing polar sphere measure, The polar surface set function on the unit sphere, Sphere integrals in angle coordinates).

$(Y_{lm})$ is a complete orthonormal family of $L^2(S^2,\sigma )$; each $V_l$ has the orthonormal basis $(Y_{l,-l},\dots ,Y_{ll})$; $V_l\perp V_k$ for $l\mathrm{\ne }k$; $H$ is dense; and the orthogonal truncations $P_{N}F$ converge to $F$ in norm (Parseval) (Sphere ONB, Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family).

Each $Y_{lm}$ has a continuous polynomial representative on $S^2$ by [F4] and Associated Legendre function normalization. For $0\le m\le l$, its chart formula is $Y_{lm}(\theta ,\phi )=b_{lm}{w}_{lm}(\cos \theta )e^{im\phi }\mathrm{/}\sqrt{2\pi }$; for $1\le m\le l$, $Y_{l,-m}=(-1{)}^m\overline{Y_{lm}}$. Thus the negative-index formula includes the factor $(-1{)}^m$, not the phase-free positive-index expression. Continuity at chart boundaries follows from the polynomial representative, and conjugation preserves it. Every $F\in H$ is consequently continuous on the whole sphere.

The restriction map ${\mathrm{r}\mathrm{e}\mathrm{s}}_l$ is a $\mathbb{C}$-linear isomorphism $H_l^{\mathrm{p}\mathrm{o}\mathrm{l}}\to V_l$, so every $F\in V_l$ has a unique representative $p_F\in H_l^{\mathrm{p}\mathrm{o}\mathrm{l}}$ with ${\mathrm{r}\mathrm{e}\mathrm{s}}_l(p_F)=F$ (Harmonic polynomial dimension (b),(c)).

Polynomial maps on ${\mathbb{R}}^3$ are $C^{\mathrm{\infty }}$; for a $C^2$ function $p$ and the linear map $L(x)=R^{-1}x$ the chain rule gives $J(p\circ L)(x)=Jp(Lx)R^{-1}$ and therefore $\mathrm{\Delta }(p\circ L)(x)=\mathrm{tr}((R^{-1}{)}^{\mathsf{T}}{D}^{2}p(Lx)R^{-1})=\mathrm{\Delta }p(Lx)$, because $R^{-1}(R^{-1}{)}^{\mathsf{T}}=I_3$; the top-degree part of $p\circ L$ is the composition of the top-degree part of $p$ with $L$, so invertibility of $L$ preserves the degree of a nonzero polynomial and preserves homogeneity; sums and scalar multiples of harmonic homogeneous polynomials of one degree are again harmonic homogeneous of that degree (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, A total derivative computes every directional derivative, and its matrix is the Jacobian, The Laplacian of a $C^2$ function and of a $C^2$ vector field, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, $C^k$ Euclidean maps and diffeomorphisms, $C^k$ Euclidean maps are closed under componentwise algebra and composition, homogeneous polynomial and homogeneous ideal, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$).

Let $\mathrm{\Phi }:S^2\to S^2$ be a measurable bijection with $\sigma ({\mathrm{\Phi }}^{-1}E)=\sigma (E)$ for all Borel $E$; then for $F,G\in L^2(S^2,\sigma )$ the functions $F\circ \mathrm{\Phi }$, $G\circ \mathrm{\Phi }$ lie in $L^2(S^2,\sigma )$, with $⟨F\circ \mathrm{\Phi },G\circ \mathrm{\Phi }⟩=⟨F,G⟩$ (substitution formula for the measure-preserving bijection), and $F=G$ $\sigma$-a.e. implies $F\circ \mathrm{\Phi }=G\circ \mathrm{\Phi }$ $\sigma$-a.e.; these identities are those of the change-of-variables formula applied to indicators and to nonnegative Borel functions and then to the four positive and negative parts (Agreement with the existing polar sphere measure, The space $L^p(\mu )$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

A continuous function on the compact metric space $S^2$ is uniformly continuous (Every continuous map from a nonempty compact Hausdorff space to a uniform space is uniformly continuous); if $g:S^2\to \mathbb{C}$ is continuous and bounded then ${\int }_{S^2}\mathrm{\mid }g{\mathrm{\mid }}^{2}d\sigma \le 4\pi \mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}^2$, because the nonnegative integral is monotone and $\sigma (S^2)=4\pi$ (The nonnegative Lebesgue integral, [F1]).

Euclidean norms on $M_3(\mathbb{R})\cong {\mathbb{R}}^9$ and on ${\mathbb{R}}^3$: for the Frobenius norm $\mathrm{\parallel }A{\mathrm{\parallel }}_F=({\sum }_{i,j}{A}_{ij}^2{)}^{1\mathrm{/}2}$, $\mathrm{\mid }Ax\mathrm{\mid }\le \mathrm{\parallel }A{\mathrm{\parallel }}_F\mathrm{\mid }x\mathrm{\mid }$ for all $x\in {\mathbb{R}}^3$ by the Cauchy-Schwarz inequality; for $R\in SO(3)$, $\mathrm{\parallel }R{\mathrm{\parallel }}_F=\sqrt{3}$ and $\mathrm{\parallel }R\mathrm{\parallel }=1$, and $R^{-1}-R_0^{-1}=R^{-1}(R_0-R)R_0^{-1}$ for $R,R_0\in SO(3)$ (Cauchy–Schwarz: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \parallel u\parallel \parallel v\parallel$, with equality exactly for linearly dependent vectors, $SO(3)$ is a subgroup of the general linear group, with $R^{-1}=R^{\mathsf{T}}$, The orthogonal group $O(3)$ and the rotation group $SO(3)$).

$SO(3)$ is a subgroup of the general linear group with $R^{-1}=R^{\mathsf{T}}$, so $(RS{)}^{-1}=S^{-1}{R}^{-1}$ and $I_3^{-1}=I_3$ for all $R,S\in SO(3)$ ($SO(3)$ is a subgroup of the general linear group, with $R^{-1}=R^{\mathsf{T}}$, The orthogonal group $O(3)$ and the rotation group $SO(3)$).