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We study here the effects of a time-dependent second order perturbation to adegenerate Ornstein-Uhlenbeck type operator whose diffusive part can be eitherlocal or non-local. More precisely, we establish that some estimates, such asthe Schauder and Sobolev ones, already known for the non-perturbed operatorstill hold, and with the same constants, when we perturb the Ornstein-Uhlenbeckoperator with second order diffusions with coefficients only depending on timein a measurable way. The aim of the current work is twofold: we weaken theassumptions required on the perturbation in the local case which has beenconsidered already in [KP17] and we extend the approach presented therein to awider class of degenerate Kolmogorov operators with non-local diffusive part ofsymmetric stable type.

About the regularity of degenerate non-local Kolmogorov operators under diffusive perturbations