Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), then for all \(x\in \mathbb R^n\), \(u_u(x)=\int _{\mathbb R^n}\prod _i p_{\sin ^2u}(\cos u\, x_i,z_i)1_A(z)\, dz\), where \(p_v(c,z)\) is the \(N(c,v)\) density.

For every \(n\ge 0\) and \(u\in \mathbb R\), the linear map \(x\mapsto \cos u\, x\) on \(\mathbb R^n\) has operator norm at most one for the maximum norm.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), \(u_u\) is differentiable at every \(x\in \mathbb R^n\), with \(D u_u(x)[h]=\cos u\int 1_A(z)\, \partial _c[\prod _i p_{\sin ^2u}(c_i,z_i)]_{c=\cos u\, x}[h]\, dz\), where \(p_v\) is the Gaussian density and \(\partial _c\) is differentiation in the mean vector.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), then \(\| D u_u(x)\| \le n(1+1/\sin ^2u)\) for every \(x\in \mathbb R^n\), with operator norm for the maximum norm.

Let \(E\) be a real normed vector space, \(D:E\to \mathcal L(E,\mathbb R)\) be \(C\)-Lipschitz in operator norm with \(C\ge 0\), and \(L:E\to E\) be continuous linear with \(\| L\| \le 1\). Then \(x\mapsto D(Lx)\circ L\) is \(C\)-Lipschitz.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), then \(x\mapsto D u_u(x)\) is globally \(2n^2/\sin ^2u\)-Lipschitz in operator norm for the maximum norm on \(\mathbb R^n\).