On $W^{2,\varepsilon}$-estimates for a class of singular-degenerate parabolic equations

Abstract: We study a class of parabolic equations in non-divergence form with measurable coefficients that are singular, degenerate, or both singular and degenerate through a weight belonging to the $A_{1+\frac{1}{n}}$ -Muckenhoupt class of weights. Under some smallness assumption on a weighted mean oscillation of the weight, F.-H. Lin type weighted $W^{2,\varepsilon}$-estimates are proved. To prove the result, we establish a result on local quantitative lower estimates of solutions to the class of equations, which are known as the mean sojourn times of sample paths within sets. This type of estimate was proved by L. C. Evans for the class of linear elliptic equations in non-divergence form with uniformly elliptic and bounded measurable coefficients. A class of weighted parabolic cylinders intrinsically suitable for the class of equations is introduced. The parabolic ABP estimates, and a perturbation method are used to overcome the singularity and degeneracy of the coefficients. Careful analysis on regularization and truncation of the weights is performed. The paper provides foundational ingredients and estimates for the study of fully nonlinear parabolic equations with singular-degenerate coefficients.