  The main result of this note is the strengthening of a quite arbitrary a
priori Fourier restriction estimate to a multi-parameter maximal estimate of
the same type. This allows us to discuss a certain multi-parameter Lebesgue
point property of Fourier transforms, which replaces Euclidean balls by
ellipsoids. Along the lines of the same proof, we also establish a
$d$-parameter Menshov--Paley--Zygmund-type theorem for the Fourier transform on
$\mathbb{R}^d$. Such a result is interesting for $d\geq2$ because, in a sharp
contrast with the one-dimensional case, the corresponding endpoint $L^2$
estimate (i.e., a Carleson-type theorem) is known to fail since the work of C.
Fefferman in 1970. Finally, we show that a Strichartz estimate for a given
homogeneous constant-coefficient linear dispersive PDE can sometimes be
strengthened to a certain pseudo-differential version.
