Let \(f:\mathbb R^n\to \mathbb R\) have derivative \(L\) at \(x\in \mathbb R^n\), and let \(y\in \mathbb R^n\). Then \(\theta \mapsto f(\cos \theta \, x+\sin \theta \, y)\) has derivative \(L(y)\) at zero.

Let \(f:\mathbb R^n\to \mathbb R\) have derivative \(L\) at \(x\), and let \(y\in \mathbb R^n\). Then \(|f(\cos \theta \, x+\sin \theta \, y)-f(x)|/\theta \to |L(y)|\) as \(\theta \to 0^+\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. For a linear functional \(L:\mathbb R^n\to \mathbb R\), put \(|L|_2=\sqrt{\sum _i L(e_i)^2}\), where \(e_i\) is the canonical basis. Then \(\mathbb E|L(X)|=|L|_2\, 2\varphi (0)\), where \(\varphi \) is the one-dimensional standard Gaussian density.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Let \(f:\mathbb R^n\to \mathbb R\) be differentiable everywhere with derivative \(L(x)\), and put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\), where \(e_i\) is the canonical basis. Let \(C\in [0,\infty ]\). If \(\mathbb E(|f(\cos \theta \, X+\sin \theta \, Y)-f(X)|/\theta )\le C\) for every \(\theta \gt 0\), then \(\mathbb E|L(X)Y|\le C\), with both expectations interpreted as nonnegative extended integrals.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Every continuous real linear functional \(L:\mathbb R^n\to \mathbb R\) is \(\gamma _n\)-integrable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. For a linear functional \(L:\mathbb R^n\to \mathbb R\), put \(|L|_2=\sqrt{\sum _i L(e_i)^2}\), where \(e_i\) is the canonical basis. The nonnegative extended expectation satisfies \(\mathbb E|L(X)|=|L|_2\, 2\varphi (0)\), with the finite expression on the right embedded into \([0,\infty ]\) and \(\varphi \) the standard Gaussian density.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. If \(L:\mathbb R^n\to \mathcal L(\mathbb R^n,\mathbb R)\) is continuous, put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\). Then \(\mathbb E|L(X)Y|=(\mathbb E|L(X)|_2)\, 2\varphi (0)\) as nonnegative extended integrals, where \(\varphi \) is the standard Gaussian density.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Let \(f:\mathbb R^n\to \mathbb R\) be differentiable everywhere with derivative \(L(x)\), and put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\), where \(e_i\) is the canonical basis. Assume \(L\) is continuous and \(C\in [0,\infty ]\). If \(\mathbb E(\sqrt{\pi /2}\, |f(\cos \theta \, X+\sin \theta \, Y)-f(X)|/\theta )\le C\) for every \(\theta \gt 0\), then \(\mathbb E|L(X)|_2\le C\), with all expectations nonnegative extended integrals.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Let \(f:\mathbb R^n\to \mathbb R\) be differentiable everywhere with derivative \(L(x)\), and put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\), where \(e_i\) is the canonical basis. Assume \(L\) is continuous, \(0\le f\le 1\), and \(C\in [0,\infty ]\). If \(\sqrt{\pi /2}\, \mathbb E|f(\cos \theta \, X+\sin \theta \, Y)-f(X)|/\theta \le C\) for every \(\theta \gt 0\), where the real expectation is embedded into \([0,\infty ]\), then the nonnegative extended expectation satisfies \(\mathbb E|L(X)|_2\le C\).