  The purpose of this paper is to improve the known estimates for Mockenhaupt's
square function in $\mathbb R^3$ and for Sogge's local smoothing in $\mathbb
R^{2+1}$ spacetime. For this we use the trilinear approach of S. Lee and A.
Vargas for the cone multiplier with some trilinear estimates obtained from the
$\ell^2$ decoupling theorem and multilinear restriction theorem.
