Nonlinear gradient estimates for parabolic problems with irregular obstacles

We establish the natural Calderón–Zygmund theory for solutions to parabolic variational inequalities satisfying an irregular obstacle constraint and involving degenerate/singular operators in divergence form of general type, and proving that the (spatial) gradient of solutions is as integrable as both the (spatial) gradient of the obstacles and the inhomogeneous terms, under the assumption that the involved nonlinearities have small a BMO semi-norm in the spatial variables while they are allowed to be merely measurable in the time variable.