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TITLE: [GS_M_APP] Weighted Sobolev Estimates for Equations with Fabes-Kenig-Seraponi Singular-Degenerate Type Coefficients

ABSTRACT: We report recent results on the quantitative regularity theory in weighted Sobolev spaces for second order linear divergence-form parabolic equations. The leading coefficients of these equations exhibit singular and degenerate behaviors of Fabes-Kenig-Seraponi type, characterized by Muckenhoupt weight classes. We begin by recalling classical foundations. Then, we turn to recent developments for elliptic equations with these singular-degenerate coefficients before introducing our new results in the parabolic setting. The proofs rely on a non-homogeneous weighted parabolic cylinder framework and a Lions-Aubin type compactness theorem. These tools are combined with the freezing-coefficient technique, a compactness argument, and the level-set method introduced by Caffarelli and Peral to derive the desired gradient estimates.