Singular integral theory plays a very important role in Harmonic analysis. It originated from partial differential equation and complex analysis. Many scholars in Harmonic analysis have excellent works in this field, such as the winner of Wolf Prize in Mathematics Calder\'on, Stein, Fefferman, the speakers of ICM (45 minutes) Christ, Lacey, Sogge et al.. Singular integral operators and their related operators, including maximal singular integral operators, strong maximal operators, commutators of singular integral operators, square functions etc., have played crucial roles in Harmonic analysis. As a nontrivial generalization of linear theory, the theory of multilinear singular integral originated from a series of famous works of Coifman and Meyer in the 70s. The study of singular integral operators also promotes the development of partial differential equation and other mathematical areas, for instance, Fabes et al. applied multilinear Littlewood-Paley function to the estimate of the square root of an elliptic operator in divergence form, and the estimate of solutions to the Cauchy problem for nondivergence-form parabolic equations. Then, it was systematically studied by Grafakos and Torres. Later, Grafakos et al. comprehensively studied the boundedness of bilinear integral operators with rough kernels. However, there are many open problems in this field, such as endpoint estimates, the sparse domination and weighted estimates for bilinear singular integral operators with rough kernels. The non-standard singular integral with rough kernels was closely related to the Calder\'on commutator and first considered by Cohen. Subsequently, Hofmann improved the result of Cohen. He weakened Cohen's assumption {\rm Lip}_{\alpha}(\mathbb{S}^{n-1})\, (0<\alpha\leq1)   to   \bigcup_{q>1}L^q(\mathbb{S}^{n-1})   and obtained the weighted estimates of   T_{\Omega,A}   when   \Omega\in L^{\infty}{(\mathbb{S}^{n-1})}  . 

   This dissertation mainly studies some unsolved problems of the above operators and their related operators. Firstly, we study the strong and weak   L\log L   endpoint estimate of the non-standard singular integrals with rough kernels. Secondly, we consider the weighted boundedness of bilinear singular integral operators with rough kernels and their associated maximal operator. Finally, we study the weighted estimates for bilinear Fourier multiplier operators with multiple weights. 

  The dissertation is arranged as follows:

  In introduction, we give some background and previous known results of the above operators. We also briefly state the main results of this dissertation. 

  In Chapter 1, we study the boundedness of the non-standard singular integral operators   T_{\Omega,\,A}   with rough kernels. More precisely, we obtain the following results: 

  1. We establish the strong   (p,p)   type and weak   L\log L   endpoint estimates for   T_{\Omega,\,A}   when either of the following condition is satisfied. (i).   \Omega\in L(\log L)^{2}(\mathbb{S}^{n-1})   and   T_{\Omega,\,A}   is bounded on   L^2(\mathbb{R}^n)  ; (ii).   \Omega\in L(\log L)^{3}(\mathbb{S}^{n-1})  . 

  2. We establish the sparse domination for   T_{\Omega,\,A}   when   \Omega\in L^{\infty}(\mathbb{S}^{n-1})  . 

  3. W give certain quantitative strong and weak   L\log L   endpoint weighted estimates for   T_{\Omega,\,A}  . To prove the above estimates, the key point is that if the kernel doesn't have enough smoothness condition, we can't obtain the boundedness estimates for   T_{\Omega,\,A}   by classical methods. It is worth noting that, the key ingredient in our proof is the estimate for the bad part in the Calder\'on-Zygmund decomposition. Then, we find all the smooth radial truncations function with kernels satisfy the size condition and smoothness condition. Additionally, we apply many techniques, such as the method of rotation of Calder\'on-Zygmund, Littlewood-Paley decomposition, microlocal decomposition, stopping collection etc. The boundedness  of   T_{\Omega,\,A}   and   \widetilde{T}_{\Omega,\,A}   enables us to get rid of self-adjoint property. Essentially, these results improve the strong   (p,p)   boundedness of Cohen and Hofmann. Meanwhile, we improve the result of Hu when   \Omega\in {\rm Lip}_{\alpha}(\Sp^{n-1})   to the case of completely rough kernels. 

   In Chapter 2, we establish the sparse domination and weighted estimates for bilinear singular integral operators   T_{\Omega}   with rough kernels when   \Omega \in L^{r}(\mathbb S^{2n-1})  ,   4/3

   In Chapter 3, we study the boundedness of bilinear maximal singular integral operators   T^*_{\Omega}   with rough kernels. Firstly, by Calder\'on-Zygmund decomposition, Littlewood-Paley decomposition, stopping collection etc., we obtain sparse dominations and some quantitative weighted estimates for   T^*_{\Omega}  . Moreover, with a pointwise sparse domination for the iterated commutators of bilinear maximal operators, we present some quantitative weak and strong type weighted estimates for these commutators. Finally, we also gain the local decay estimate, the Coifman-Fefferman inequality with   w\in A_{\infty}, and the Fefferman-Stein inequality with arbitrary weights for the iterated commutators. In essence, we extend the results of Barron to bilinear maximal singular integral operators. 

   In Chapter 4, we establish the weighted estimate for bilinear Fourier multiplier operators with multiple weights. In 2013, Miyachi and Tomita gave the minimal smoothness condition to guarantee the   (L^{p_1}\times L^{p_2}, L^{p})   boundedness and   (H^{p_1}\times H^{p_2}, L^p)     (p\geq 2/3)   boundedness of the bilinear Fourier multiplier to be held. We will consider the weighted estimate for   T_{\sigma}   when   \sigma   satisfies the minimal smoothness condition with more general multiple weights   \vec{w}\in A_{\vec{p}/\vec{r}}(\mathbb{R}^{2n})  . The novelty of our proof is the sparse domination of   T_{\sigma}  . Essentially, we extend the product weights estimates of Miyachi and Tomita to more general multiple weights.