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In this paper the authors consider the weighted estimates for the Calderón commutator defined bywith P2(A;x, y) = A(x) − A(y) − A′(y)(x − y) and A′ ∈ BMO(ℝ). Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from 
\mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y,
$L^{p_1}(\mathbb {R},w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R},w_{m+1})$ to 
$L^{p}(\mathbb {R},\nu _{\vec {\kern 1pt w}})$, with p1, …, pm+1 ∈ (1, ∞), 1/p = 1/p1 + · · · + 1/pm+1, and 
$\vec {\kern 1pt w}=(w_1, \ldots , w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$. The authors also obtain the weighted weak type endpoint estimates for 
$\mathcal {C}_{m+1, A}$.
In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calderón commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the 
$L^{p}(\mathbb{R}^{d},w)$ space, including some peculiar endpoint estimates of the higher dimensional Calderón commutator.
In this paper, we prove that the singular integral defined by
${{T}_{\Omega ,a}}f(x)=\text{p}\text{.}\text{v}\text{.}{{\int }_{{{\mathbb{R}}^{d}}}}\frac{\Omega (x-y)}{|x-y{{|}^{d}}}\cdot {{m}_{x,y}}a\cdot f(y)dy$is bounded on 
${{L}^{p}}({{\mathbb{R}}^{d}})$ for 
$1\,<\,p\,<\,\infty $ and is of weak type (1,1), where 
$\Omega \,\in L\text{lo}{{\text{g}}^{+}}L({{S}^{d-1}})$ and 
${{m}_{x,y}}a\,=:\,\int{_{0}^{1}}\,a(sx\,+\,(1\,-\,s)y)ds$, with 
$a\,\in \,{{L}^{\infty }}({{\mathbb{R}}^{d}})\,$ satisfying some restricted conditions.
