Abstract
Let be a positive Radon measure on which may be nondoubling. The only condition that satisfies is for all , , and some fixed constant . In this paper, we introduce the operator related to such a measure and assume it is bounded on . We then establish its boundedness, respectively, from the Lebesgue space to the weak Lebesgue space , from the Hardy space to and from the Lesesgue space to the space . As a corollary, we obtain the boundedness of in the Lebesgue space with .
Highlights
A positive Radon measure μ on Rd is said to be doubling if there exists some constant C such that μ B x, 2r ≤ Cμ B x, r for all x ∈ supp μ, r > 0
In the recent years, it has been shown that a big part of the classical theory remains valid if the doubling assumption on μ is substituted by the growth condition as follows: μ B x, r ≤ C0rn
In 2001, Tolsa in 1, 2 investigated the weak 1,1 inequality for singular integrals, the LittlewoodPaley theory and the T 1 theorem with nondoubling measures
Summary
A positive Radon measure μ on Rd is said to be doubling if there exists some constant C such that μ B x, 2r ≤ Cμ B x, r for all x ∈ supp μ , r > 0. Let ψ be a function on Rd such that there exist positive constants C0, C1, δ, and γ satisfying a ψ ∈ L1 Rd and Rd ψ x dμ x 0, b |ψ x | ≤ C0 1 |x| −n−δ, c |ψ x y − ψ x | ≤ C1|y|γ 1 |x| −n−γ−δ for 2|y| ≤ |x| For this ψ, we define the Littlewood-Paley’s gλ∗,μ-function with nondoubling measures as follows: gλ∗,μ f x. A function f ∈ L1loc μ is said to be in the space RBMO μ if there exists some constant C > 0 such that for any cube Q centered at some point of supp μ. A function f ∈ L1loc μ positive constant C such that for any is4s√adid,to4√bedinn the space RBLO μ if there exists 1 doubling cube Q, some mQ f. A B will always denote that there exists a constant C > 0, such that A ≤ CB
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