This paper is devoted to the study of
$P$
-regularity of viscosity solutions
$u(x,P)$
,
$P\in {\Bbb R}^n$
, of a smooth Tonelli Lagrangian
$L:T {\Bbb T}^n \rightarrow {\Bbb R}$
characterized by the cell equation
$H(x,P+D_xu(x,P))=\overline {H}(P)$
, where
$H: T^* {\Bbb T}^n\rightarrow {\Bbb R}$
denotes the Hamiltonian associated with
$L$
and
$\overline {H}$
is the effective Hamiltonian. We show that if
$P_0$
corresponds to a quasi-periodic invariant torus with a non-resonant frequency, then
$D_xu(x,P)$
is uniformly Hölder continuous in
$P$
at
$P_0$
with Hölder exponent arbitrarily close to
$1$
, and if both
$H$
and the torus are real analytic and the frequency vector of the torus is Diophantine, then
$D_xu(x,P)$
is uniformly Lipschitz continuous in
$P$
at
$P_0$
, i.e., there is a constant
$C\gt 0$
such that
$\|D_xu(\cdot ,P)-D_xu(\cdot ,P_0)\|_{\infty }\le C\|P-P_0\|$
for
$\|P-P_0\|\ll 1$
. Similar P-regularity of the Peierls barriers associated with
$L(x,v)- \langle P,v \rangle $
is also obtained.