Weighted \(W^{1,2}_{p(\cdot )}\)-Estimate for Fully Nonlinear Parabolic Equations with a Relaxed Convexity

We devote this paper to global estimate in weighted variable exponent Sobolev spaces for fully nonlinear parabolic equations under a relaxed convexity condition. It is assumed that the associated variable exponent is log-Hölder continuous, the weight belongs to certain Muckenhoupt class concerning the variable exponent, the leading part of nonlinearity satisfies a relaxed convexity in Hessian and is of VMO condition in space-time variables, and the boundary of underlying domain satisfies \(C^{1,1}\) $C^{1,1}$ -smooth. Our key strategy is to utilize a unified approach based on the generalized versions of Fefferman–Stein theorem of the sharp functions and extrapolation to establish the estimates of \(D^{2}u\) $D^{2}u$ and \(D_{t} u\) $D_{t}u$ within the framework of weighted variable exponent Lebesgue spaces.