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We establish new approximation results, in the sense of Lusin, of Sobolevfunctions by Lipschitz ones, in some classes of non-doubling metric measurestructures. Our proof technique relies upon estimates for heat semigroups andapplies to Gaussian and $RCD(K, \infty)$ spaces. As a consequence, we obtainquantitative stability for regular Lagrangian flows in Gaussian settings.

Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and $RCD(K,\infty)$ spaces