Baode Li

Anisotropic singular integrals in product spaces

AbstractIn this paper, the authors introduce a class of product anisotropic singular integral operators, whose kernels are adapted to the action of a pair

A Mean Characterization of Weighted Anisotropic Besov and Triebel-Lizorkin Spaces

\vec A

Weighted anisotropic product Hardy spaces and boundedness of sublinear operators

:= (A1, A2) of expansive dilations on ℝn and ℝm, respectively. This class is a generalization of product singular integrals with convolution kernels introduced in the isotropic setting by Fefferman and Stein. The authors establish the boundedness of these operators in weighted Lebesgue and Hardy spaces with weights in product A∞ Muckenhoupt weights on ℝn × ℝm. These results are new even in the unweighted setting for product anisotropic Hardy spaces.

Littlewood-Paley characterization and duality of weighted anisotropic product Hardy spaces

Let φ : ℝn × [0, ∞)→[0, ∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H A φ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations of H A φ(ℝn) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space H A p(ℝn) with p ∈ (0,1] and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization of H A φ(ℝn), and, as an application, the authors prove that, for a given admissible triplet (φ, q, s), if T is a sublinear operator and maps all (φ, q, s)-atoms with q < ∞ (or all continuous (φ, q, s)-atoms with q = ∞) into uniformly bounded elements of some quasi-Banach spaces ℬ, then T uniquely extends to a bounded sublinear operator from H A φ(ℝn) to ℬ. These results are new even for anisotropic Orlicz-Hardy spaces on ℝn.

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

In this article, the authors study weighted anisotropic Besov and TriebelLizorkin spaces associated with expansive dilations and A∞ weights. The authors show that elements of these spaces are locally integrable when the smoothness parameter α is positive. The authors also characterize these spaces for small values of α in terms of a mean square function recently introduced in the context of Sobolev spaces in [Math. Ann. 354 (2012), 589-626] and isotropic Triebel-Lizorkin spaces in [Publ. Mat. 57 (2013), 57-82].