Let $\mu$ be a zero mean Gaussian probability measure on $\mathbb{R}^n$ whose covariance is less than the identity. If $f$ is a $1$-Lipschitz real function on $\mathbb{R}^n$ such that there exists a radial vector field $v$ with:
$$ ||\nabla f - v||_{L^2(\mu)} \leq \varepsilon $$
Is it true that there exists a radial function $g$ such that:
$$ ||f - g||_{L^2(\mu)} \leq O(\varepsilon) $$
Because of the covariance assumption, $\mu$ satisfies the usual Poincare inequality. But the radial form above, while plausible, doesn't seem to follow directly from it.
