  In this paper, we investigate the multilinear boundedness properties of the
higher ($n$-th) order Calder\'on commutator for dimensions larger than two. We
establish all multilinear endpoint estimates for the target space
$L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, including that Calder\'on commutator
maps the product of Lorentz spaces $L^{d,1}(\mathbb{R}^d)\times\cdots\times
L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to
$L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, which is the higher dimensional
nontrivial generalization of the endpoint estimate that the $n$-th order
Calder\'on commutator maps $L^{1}(\mathbb{R})\times\cdots\times
L^{1}(\mathbb{R})\times L^1(\mathbb{R})$ to
$L^{\frac{1}{1+n},\infty}(\mathbb{R})$. When considering the target space
$L^{r}(\mathbb{R}^d)$ with $r<\frac{d}{d+n}$, some counterexamples are given to
show that these multilinear estimates may not hold. The method in the present
paper seems to have a wide range of applications and it can be applied to
establish the similar results for Calder\'on commutator with a rough
homogeneous kernel.

Classical Analysis and ODEs