  In this paper, we study parabolic equations in divergence form with
coefficients that are singular degenerate as some Muckenhoupt weight functions
in one spatial variable. Under certain conditions, weighted reverse
H\"{o}lder's inequalities are established. Lipschitz estimates for weak
solutions are proved for homogeneous equations with singular degenerate
coefficients depending only on one spatial variable. These estimates are then
used to establish interior, boundary, and global weighted estimates of
Calder\'{o}n-Zygmund type for weak solutions, assuming that the coefficients
are partially VMO (vanishing mean oscillations) with respect to the considered
weights. The solvability in weighted Sobolev spaces is also achieved. Our
results are new even for elliptic equations, and non-trivially extend known
results for uniformly elliptic and parabolic equations. The results are also
useful in the study of fractional elliptic and parabolic equations with
measurable coefficients.

Analysis of PDEs