Abstract
Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts {Πj}j=13 defined via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts defined via regular wavelets. Precisely, the author first obtains the boundedness of Π3 on Hardy spaces and then, via the methods of interpolation and the well-known T(1) theorem, establishes the endpoint estimates for {Πj}j=13. The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of {Πj}j=13, which has independent interests. It is also remarked that, throughout this article, μ is not assumed to satisfy the reverse doubling condition.
Highlights
Via the methods of interpolation and the crucial estimates (11) and (12), we show that K f has the weak boundedness property WBP(η ) with η as in Lemma 2 below
We mainly state some preliminary notions and results which are needed to the proof of the main results Theorems 2–4 below
We investigate some relations among the boundedness of {Π j }3j=1
Summary
The triple (X , d, μ) is called a space of homogeneous type if μ is a non-negative measure satisfying the following doubling condition: there exists a positive constant C(X ) ∈ [1, ∞), depending on X , such that, for any r ∈ (0, ∞) and x ∈ X , creativecommons.org/licenses/by/ A sequence { Qk }k∈Z of bounded linear integral operators on L2 (X ) is called an approximation of the identity with exponential decay (for short, exp-ATI) if there exist constants C, ν ∈ (0, ∞), a ∈ (0, 1] and η ∈ (0, 1) such that, for any k ∈ Z, the kernel of the operator Qk , which is still denoted by Qk , satisfies (i)
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