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

Abstract

In this paper we establish a discrete Calderón's identity which converges in both L q (R n+m ) (1 < q < ∞) and Hardy space H p (R n × R m ) (0 < p ≤ 1). Based on this identity, we derive a new atomic decomposition into (p, q)-atoms (1 < q < ∞) on H p (R n × R m ) for 0 < p ≤ 1. As an application, we prove that an operator T , which is bounded on L q (R n+m ) for some 1 < q < ∞, is bounded from H p (R n ×R m )

Differentiation of the integral, Hardy spaces and Calderón–Zygmund operators in the product setting

“…The fact that, for p = 1, Chang-Fefferman elementary particles do not need to be smooth nor in L ∞ is known [22]. For 0 < p ≤ 1, it was proved [16] that if a function belongs to H p (R n 1 × R n 2 ) ∩ L q (R n 1 +n 2 ), for some 1 < q < ∞, then it can be written as a sum of (p, q)-product-atoms with coefficients in l p , where the so called "(p, q)-product-atoms" [16] are rough (p, q)-atoms requiring some extra conditions. Unlike the results in [16], which depend on a discrete Calderón's identity, ours rely on variants of Journé's lemma.…”

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

“…Thus, deriving an atomic decomposition in the product Hardy spaces by using (p, q)-atoms for q = 2 becomes interesting. This has been recently carried out in [14] by using discrete Littlewood-Paley analysis and the discrete Calderón's identity.…”

Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition

“…In the multi-parameter situations, atomic [1][2][3]10], and was applied to prove the boundedness of multi-parameter singular integral operators on Hardy spaces, using the atomic decomposition together with the Journé's covering lemma, by Fe erman [11], Journé [25,26], Pipher [33], Han, Lu and Ruan [21,22] etc. A more re ned and improved version of atomic decomposition in the multi-parameter Hardy spaces was carried out in [24] and a boundedness criterion was established using the atomic decomposition. Therefore, it is interesting and useful to establish the atomic decomposition and the boundedness criterion on the multi-parameter Triebel-Lizorkin spaces associated with the composition of two dilations of di erent homogeneities.…”

“…Firstly, when 0 < ≤ 1, = 2, = (0, 0), we then obtain the definition of atoms in com . Di erent from classical de nitions of atoms in pure product Hardy spaces [11,24], an additional good condition: ‖ ‖ com ≤ 0 is involved. This condition can be obtained from the remaining conditions in pure product spaces (see the appendix of [24]).…”