  This paper studies the Sobolev regularity of weak solution of degenerate
elliptic equations in divergence form $\text{div}[\mathbf{A}(X) \nabla u] =
\text{div}[\mathbf{F}(X)]$, where $X = (x,y) \in \mathbb{R}^{n} \times
\mathbb{R}$ . The coefficient matrix $\mathbf{A}(X)$ is a symmetric, measurable
$(n+1) \times (n+1)$ matrix, and it could be degenerate or singular in the one
dimensional $y$-variable as a weight function in the Muckenhoupt class $A_2$ of
weights. Our results give weighted Sobolev regularity estimates of
Calder\'{o}n-Zygmund type for weak solutions of this class of singular,
degenerate equations. As an application of these estimates, we establish global
Sobolev regularity estimates for solutions of the spectral fractional elliptic
equation with measurable coefficients. This result can be considered as the
Sobolev counterpart of the recently established Schauder regularity theory of
fractional elliptic equations.
