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

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