  We investigate $L^p$ boundedness of the maximal function defined by the
averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of
tori $\mathbb{T}_t^{s}:=\{ (
(t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\, s\sin\theta ): \theta,
\phi \in [0,2\pi) \}$ with $c_0t>s>0$ for some $c_0\in (0,1)$. We prove that
the associated (two-parameter) maximal function is bounded on $L^p$ if and only
if $p>2$. We also obtain $L^p$--$L^q$ estimates for the local maximal operator
on a sharp range of $p,q$. Furthermore, the sharp smoothing estimates are
proved including the sharp local smoothing estimates for the operators $f\to
\mathcal A_t^s f$ and $f\to \mathcal A_t^{c_0t} f$. For the purpose, we make
use of Bourgain--Demeter's decoupling inequality for the cone and
Guth--Wang--Zhang's local smoothing estimates for the $2$ dimensional wave
operator.

Classical Analysis and ODEs