Pointwise Conditions for Membership of Functions in Weighted Sobolev Classes

According to a known characterization, a function $f$ belongs to the Sobolev space $W^{p,1}(\mathbb{R}^n)$ of functions contained in $L^p(\mathbb{R}^n)$ along with their generalized first-order derivatives precisely when there is a function $g\in L^p(\mathbb{R}^n)$ such that
$$ |f(x)-f(y)|\le |x-y|(g(x)+g(y)) $$
for almost all pairs $(x,y)$. An analogue of this estimate is also known for functions from the Gaussian Sobolev space $W^{p,1}(\gamma)$ in infinite dimension. In this paper the converse is proved; moreover, it is shown that the above inequality implies membership in appropriate Sobolev spaces for a large class of measures on finite-dimensional and infinite-dimensional spaces.