Regularity and invariance of the cutoff Gaussian-mixture class

By [F3], diagonalize $\mathrm{\Sigma }$ in an orthonormal basis with eigenvalues $s_1^2,s_2^2,s_3^2\in [a,b]$. The linear substitution $y_i=(v-u{)}_i\mathrm{/}{s}_i$ has Jacobian $\det (\mathrm{\Sigma }{)}^{-1\mathrm{/}2}$; [F4] in each coordinate gives $\int {\gamma }_{\mathrm{\Sigma },u}(v)dv=1$. The same eigenvalue bounds give $(\det \mathrm{\Sigma }{)}^{-1\mathrm{/}2}\in [b^{-3\mathrm{/}2},a^{-3\mathrm{/}2}]$ and $\mathrm{\mid }v-u{\mathrm{\mid }}^2\mathrm{/}b\le (v-u{)}^{\mathsf{T}}{\mathrm{\Sigma }}^{-1}(v-u)\le \mathrm{\mid }v-u{\mathrm{\mid }}^2\mathrm{/}a$, proving the two Gaussian envelopes for each component and hence for every finite mixture.

Let $O_n:{\mathbb{R}}^6\to {\mathbb{R}}^6$ be the orthogonal pair map from [F2], and let $P:{\mathbb{R}}^6\to {\mathbb{R}}^3$ project onto the first component. For component densities ${\gamma }_{\mathrm{\Sigma },u}$ and ${\gamma }_{\mathrm{T},u}$, the joint centered covariance is $S=\mathrm{diag}(\mathrm{\Sigma },\mathrm{T})$. Using [F6] under $O_n$, the density of the pair after reflection is the centered Gaussian with covariance $S_n=O_{n}S{O}_n^{\mathsf{T}}$ and mean $(u,u)$. Write $S_n$ in blocks as $\left(\begin{array}{cc}C& D\\ D^{\mathsf{T}}& E\end{array}\right)$. Its first marginal is Gaussian with covariance $C$: indeed $C$ is positive definite, the Schur complement $H=E-D^{\mathsf{T}}{C}^{-1}D$ is positive definite, and block elimination gives $\det S_n=\det C\det H$ and $(x,y{)}^{\mathsf{T}}{S}_n^{-1}(x,y)=x^{\mathsf{T}}{C}^{-1}x+(y-D^{\mathsf{T}}{C}^{-1}x{)}^{\mathsf{T}}{H}^{-1}(y-D^{\mathsf{T}}{C}^{-1}x)\mathrm{.}$ Integrating in $y$ using [F3], [F4], and [F7] gives the normalized marginal ${\gamma }_{C,u}$. Also $C=P{S}_n{P}^{\mathsf{T}}$ satisfies $aI\le C\le bI$: for every $z$, $z^{\mathsf{T}}Cz=(z,0{)}^{\mathsf{T}}{O}_{n}S{O}_n^{\mathsf{T}}(z,0)$ lies between $a\mathrm{\mid }z{\mathrm{\mid }}^2$ and $b\mathrm{\mid }z{\mathrm{\mid }}^2$.

Differentiating a component ${\gamma }_{\mathrm{\Sigma },u}$ in $v$ multiplies it by a polynomial in ${\mathrm{\Sigma }}^{-1}(v-u)$; each coefficient is uniformly bounded because $\mathrm{\parallel }{\mathrm{\Sigma }}^{-1}\mathrm{\parallel }\le a^{-1}$. The exponent has the upper bound from step 1.1, so every derivative of a finite mixture has the stated polynomial-Gaussian envelope, uniformly over all finite mixtures in this class.

Therefore $\mathcal{G}({\gamma }_{\mathrm{\Sigma },u},{\gamma }_{\mathrm{T},u})$ is the pointwise angular average ${\int }_{S^2}{\gamma }_{C_n,u}d\sigma (n)$ with $C_n\in K_{a,b}$. The map $n\mapsto C_n$ is continuous, and $C\mapsto {\gamma }_{C,u}$ is continuous in $X$: pointwise convergence follows from the formula, while the weighted upper Gaussian envelope from step 1.1 is an integrable dominator and [F5] gives weighted-$L^1$ convergence. For each $n$, continuity supplies a relative-open neighborhood on which the $X$-distance from ${\gamma }_{C_n,u}$ is less than $\epsilon$; compactness of $S^2$ gives a finite subcover. Successively subtracting earlier open sets makes this a finite Borel partition $E_j$, and choosing one $n_j$ in each nonempty piece gives variation at most $2\epsilon$ on that piece. Tonelli [F7] and the triangle inequality bound the weighted-$L^1$ distance from the angular average to ${\sum }_j\sigma (E_j){\gamma }_{C_{n_j},u}$ by $2\epsilon {\sum }_j\sigma (E_j)=2\epsilon$. Since $\epsilon >0$ is arbitrary, the angular average belongs to ${\mathcal{M}}_{u,a,b}$. Bilinear expansion proves the same for finite convex mixtures in both inputs.

For general $f,g\in {\mathcal{M}}_{u,a,b}$, choose finite-mixture sequences $f_k\to f$ and $g_k\to g$ in $X$. The bound [F2] gives $\mathrm{\parallel }\mathcal{G}(f_k,g_k)-\mathcal{G}(f,g){\mathrm{\parallel }}_{1,w_0}\le 2\mathrm{\parallel }{f}_k-f{\mathrm{\parallel }}_{1,w_0}\mathrm{\parallel }{g}_k{\mathrm{\parallel }}_{1,w_0}+2\mathrm{\parallel }f{\mathrm{\parallel }}_{1,w_0}\mathrm{\parallel }{g}_k-g{\mathrm{\parallel }}_{1,w_0}⟶0.$ Each $\mathcal{G}(f_k,g_k)$ is in ${\mathcal{M}}_{u,a,b}$ by step 2.2, so closedness proves $\mathcal{G}(f,g)\in {\mathcal{M}}_{u,a,b}$.

Let $f_k$ be finite mixtures converging in $X$ to $f\in {\mathcal{M}}_{u,a,b}$. By step 2.1, every derivative sequence $({\mathrm{\partial }}^{\alpha }{f}_k{)}_k$ is uniformly bounded and equicontinuous on every closed ball, since the next derivatives have a common bound there. On the countable dense rational grid in each ball, bounded real sequences admit convergent subsequences: repeatedly bisect a bounded interval, retain a half containing infinitely many terms, and use the nested intervals of lengths tending to zero to select a convergent subsequence. A countable diagonal selection over grid points, integer-radius balls, and multi-indices makes every derivative converge on every grid. Equicontinuity and a finite rational $\delta$-net of each ball then make these subsequences uniformly Cauchy there, hence locally uniformly convergent. The fundamental theorem of calculus along coordinate segments identifies the limits as derivatives of a smooth function $\tilde{f}$. Weighted-$L^1$ convergence identifies $\tilde{f}=f$ almost everywhere on every ball, so $\tilde{f}$ represents $f$. The pointwise envelopes and derivative bounds pass to the local uniform limit. This proves part 1 for every class element.