A trilinear approach to square function and local smoothing estimates for the wave operator

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 ${\mathrm{\ell }}^2$ decoupling theorem and multilinear restriction theorem.