Weak factorization of the Hardy space \(H^p\) for small values of \(p\), in the multilinear setting (Q2302910)

As well-known, \textit{R. R. Coifman} et al. [Ann. Math. (2) 103, 611--635 (1976; Zbl 0326.32011)] first presented a weak factorization of the Hardy space \(H^1(\mathbb{R}^n)\) through the boundedness of commutators of the Calderón-Zygmund operator. Subsequently, \textit{A. Uchiyama} [Pac. J. Math. 92, 453--468 (1981; Zbl 0493.42032)] gave a weak factorization of Hardy spaces \(H^p(X)\) in the space of homogeneous type for certain \(p\le1\) in term of Calderón-Zygmund operators, without assuming any boundedness of the commutators. In [Can. Math. Bull. 60, No. 3, 571--585 (2017; Zbl 1372.42018)], \textit{J. Li} and \textit{B. D. Wick} adopted Uchiyama's method to show a weak factorization of \(H^1(\mathbb{R}^n)\) in term of the multilinear Riesz transforms. In this paper, the author gives a weak factorization of Hardy spaces \(H^p(\mathbb{R}^n)\) for \(n/(n+1)< p <1 \) in term of multilinear Calderón-Zygmund operators. As an application, a new characterization of the Lipschitz spaces \(\mathrm{Lip}_\alpha(\mathbb{R}^n)\) via commutators of the multilinear Calderón-Zygmund operators is established.