Pointwise characterizations of Sobolev functions

We shall discuss relatively new characterizations of functions $f$ in Sobolev classes with order of integrability $p>1$ by means of inequalities of the form
$$|f(x)-f(y)| \le |x-y| [g(x)+g(y)]$$
with functions $g$ from $L^p$. We shall discuss such estimates in case of weighted Sobolev classes and Sobolev classes on infinite-dimensional spaces with measures. Here the questions arise about necessity and about sufficiency of such estimates for membership in Sobolev classes.