Combinatorics

Maximal function

Mathematics

Dual norm

Banach space

Sublinear function

Standard probability space

Mathematical analysis

Function space

Political science

Norm (philosophy)

Lp space

Pointwise

Hardy space

Communications on Pure &amp Applied Analysis

Let $ \vec{p}\in(0, \infty)^n $ and $ A $ be a general expansive matrix on $ \mathbb{R}^n $. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $ H_A^{\vec{p}}(\mathbb{R}^n) $ associated with $ A $ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $ H_A^{\vec{p}}(\mathbb{R}^n) $, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood–Paley $ g $-functions or $ g_{\lambda}^\ast $-functions via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. In addition, the authors also obtain the duality between $ H_A^{\vec{p}}(\mathbb{R}^n) $ and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from $ H_A^{\vec{p}}(\mathbb{R}^n) $ into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional $ \delta $-type and non-convolutional $ \beta $-order Calderón–Zygmund operators from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to itself [or to $ L^{\vec{p}}(\mathbb{R}^n) $]. As a corollary, the boundedness of anisotropic convolutional $ \delta $-type Calderón–Zygmund operators on the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $ with $ \vec{p}\in(1, \infty)^n $ is also presented.

Real-variable characterizations of new anisotropic mixed-norm Hardy spaces