In this work, we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) the doubly truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited smoothness. The a.e. convergence is deduced from the $L^2\times \cdots \times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.

Let $\Omega $ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$ , $T_{\Omega }$ be the convolution singular integral operator with kernel $\frac {\Omega (x)}{|x|^d}$ . In this paper, we prove that if $\Omega \in L\log L(S^{d-1})$ , and U is an operator which is bounded on $L^2(\mathbb {R}^d)$ and satisfies the weak type endpoint estimate of $L(\log L)^{\beta }$ type, then the composition operator $UT_{\Omega }$ satisfies a weak type endpoint estimate of $L(\log L)^{\beta +1}$ type.