  Denote by $\Delta$ the Laplacian and by $\Delta_\infty $ the
$\infty$-Laplacian. A fundamental inequality is proved for the algebraic
structure of $\Delta v\Delta_\infty v$: for every $v\in C^\infty$, $$\ | {
|D^2vDv|^2} - {\Delta v \Delta_\infty v } -\frac12[|D^2v|^2-(\Delta
v)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on
this, we prove the following results:
  1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have
$$|Du|^{\frac{p-\gamma}2}Du\in W^{1,2}_{\rm loc},$$ with
$\gamma<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when
$p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the known $W^{2,q}_{\rm
loc}$-regularity of $p$-harmonic functions for some $q>2$.
  2. When $n\ge 2$ and $p\in(1,2)\cup(2,3+\frac2{n-2})$, the viscosity
solutions to parabolic normalized $p $-Laplace equation have the $W_{\rm
loc}^{2,q}$-regularity in the spatial variable and the $W_{\rm
loc}^{1,q}$-regularity in the time variable for some $q>2$. Especially, when
$n=2$ an open question in [17] is completely answered.
  3. When $n\ge 1 $ and $p\in(1,2)\cup(2,3)$, the weak/viscosity solutions to
parabolic $p $-Laplace equation have the $W_{\rm loc}^{2,2}$-regularity in the
spatial variable and the $W_{\rm loc}^{1,2}$-regularity in the time variable.
The range of $p$ (including $p=2$ from the classical result) here is sharp for
the $W_{\rm loc}^{2,2}$-regularity.
