Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type

Abstract

This paper is inspired by the work of Nagel and Stein in which the L p (1 < p < ∞) theory has been developed in the setting of the product Carnot-Carathéodory spaces M = M 1 × • • • × M n formed by vector fields satisfying Hörmander's finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product BM O space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.

Pointwise multipliers of Besov spaces on Carnot–Carathéodory spaces

“…Since there are no Fourier transforms on spaces of homogeneous type, the proof of Theorem 1.9 is quite different from the proof of Theorem 2.8.2 in [8]. The key new ingredient in the proof of Theorem 1.9 is to apply the following discrete Calderón reproducing formula established in [12,10]. This formula can be stated as follows.…”

“…The real vector fields {X 1 , X 2 } and their commutators of order ≤ m span the tangent space to X at each point. See [12,10] for more details and references therein.…”

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

“…The first two authors have treated the Euclidean flag structure in [Han and Lu 2008]; see also the multiparameter setting associated with the Zygmund dilation [Han and Lu 2010]. The ideas developed in this paper and [Han and Lu 2008;Han and Lu 2010] have been adapted to some other multiparameter cases, such as the product spaces of Carnot-Carathéodory spaces [Han et al 2013a], where the L p theory was established in [Nagel and Stein 2004], and the composition of two singular integrals with different homogeneity [Han et al 2013b]. …”

Discrete Calderón’s Identity, Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

“…This general philosophy in the product spaces of homogeneous type has also been established in [17]. The duality theory for multiparameter Hardy spaces has been established in [16] using the idea of discretization.…”