Hessian estimates for equations involving p-Laplacian via a fundamental inequality

In this paper, we prove a fundamental inequality for the algebraic structure of ΔvΔ_∞v: for every v∈C^∞, [Formula presented] where Δ is the Laplacian and Δ_∞ is the ∞-Laplacian. Based on this, we prove the following results: (i) For any p-harmonic functions u with p∈(1,2)∪(2,∞), we have [Formula presented] As a by-product, when [Formula presented], we give a new proof of the known W_loc ^2,q-regularity of p-harmonic functions for some q>2. (ii) When n≥2 and [Formula presented], the viscosity solutions to the parabolic normalized p-Laplace equation have the W_loc ^2,q-regularity in the spatial variable and the W_loc ^1,q-regularity in the time variable for some q>2. Especially, when n=2 an open question in [18] is completely answered. (iii) When n≥1 and p∈(1,2)∪(2,3), the weak or viscosity solutions to the parabolic p-Laplace equation have the W_loc ^2,2-regularity in the spatial variable and the W_loc ^1,2-regularity in the time variable. The range of p here (including p=2 from the classical result) is sharp for the W_loc ^2,2-regularity.