Second order Sobolev regularity for normalized parabolic p (x) -Laplace equations via the algebraic structure

Denote by the Laplacian and by _ the -Laplacian. A fundamental inequality is proved for the algebraic structure of v_ v: for every v C^, | |D²vDv|²- v_ v-12|D²v|²- (v) ²|Dv|²| -22|D²v|²|Dv|²-|D²vDv|² Based on this, we prove the result: When n2 and p (x) (1, 2) (2, 3+2n-2), the viscosity solutions to parabolic normalized p (x) -Laplace equation have the W^2, 2₋₎₂-regularity in the spatial variable and the W^1, 2₋₎₂-regularity in the time variable.