Lusin-type theorems on the Wiener space

There are a number of classical results on the approximation of Sobolev functions by Lipschitz continuous mappings on Euclidean spaces in the sense of Lusin, as well as Lusin-type approximations of measurable vector fields by gradients of Lipschitz-continuous functions. However, the classical proofs rely on the doubling property of the Lebesgue measure and some other techniques which are specific to the finite-dimensional setting.

Nevertheless, it turns out that for a infinite-dimensional space equipped with a Gaussian measure there are natural counterparts of these results, although some questions remain open. We will discuss some new observations in this area based on the estimates for heat semigroups and some dimension-independent bounds.