Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces |
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Authors: | Qi-quan Fang Xian-liang Shi |
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Affiliation: | College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081,China |
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Abstract: | In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ
Ω
ρ
, μ
Ω,λ
*,ρ
and μ
Ω,S
ρ
on BMO(ℝ
n
) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog+
L(S
n−1). Here μ
Ω,λ
*,ρ
and μ
Ω,S
ρ
are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g
λ
*-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ
n
) or to a certain Campanato space, then μ
Ω,λ
*,ρ
(f)]2, μ
Ω,S
ρ
(f)]2 and μ
Ω
ρ
(f)]2 are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established. |
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Keywords: | parametric Marcinkiewicz integrals Campanato space weak Dini-type regularity condition |
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