About the Regularity of Degenerate Non-local Kolmogorov Operators Under Diffusive Perturbations

We study here the effects of a time-dependent second order perturbation
to a degenerate Ornstein-Uhlenbeck type operator whose diffusive part can be either local or non-local. More precisely, we establish that some estimates, such as the Schauder and Sobolev ones, already known for the non-perturbed operator still hold, and with the same constants, when we perturb the Ornstein-Uhlenbeck operator with second order diffusions with coefficients only depending on time in a measurable way. The aim of the current work is two-fold: we weaken the assumptions required
on the perturbation in the local case which has been considered already in Marino et al. (Stud Math 267(3):321–346, 2022) and we extend the approach presented therein to a wider class of degenerate Kolmogorov operators with non-local diffusive part of symmetric stable type.