Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces

In this article, we consider the Marcinkiewicz integrals with variable kernels defined by μΩ(f)(x)=(∫0∞|∫|x−y|≤tΩ(x,x−y)|x−y|n−1f(y)dy|2dtt3)1/2, where Ω(x,z)∈L∞(ℝn)×Lq(Sn−1) for q > 1. We prove that the operator μΩ is bounded from Hardy space, Hp(ℝn), to Lp(ℝn) space; and is bounded from weak Ha...

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Main Authors: Xiangxing Tao, Xiao Yu, Songyan Zhang
Format: Article
Language:English
Published: Hindawi Limited 2010-01-01
Series:Journal of Function Spaces and Applications
Online Access:http://dx.doi.org/10.1155/2010/271905
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spelling doaj-c12c3cc9add94153a9b531c51086cd142020-11-25T01:09:25ZengHindawi LimitedJournal of Function Spaces and Applications0972-68022010-01-018111610.1155/2010/271905Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spacesXiangxing Tao0Xiao Yu1Songyan Zhang2Department of Mathematics, Zhejiang University of Science & Technology, Hangzhou, Zhejiang province, 310023, ChinaDepartment of Mathematics, Zhejiang University of Science & Technology, Hangzhou, Zhejiang province, 310023, ChinaDepartment of Mathematics, Zhejiang University of Science & Technology, Hangzhou, Zhejiang province, 310023, ChinaIn this article, we consider the Marcinkiewicz integrals with variable kernels defined by μΩ(f)(x)=(∫0∞|∫|x−y|≤tΩ(x,x−y)|x−y|n−1f(y)dy|2dtt3)1/2, where Ω(x,z)∈L∞(ℝn)×Lq(Sn−1) for q > 1. We prove that the operator μΩ is bounded from Hardy space, Hp(ℝn), to Lp(ℝn) space; and is bounded from weak Hardy space, Hp,∞(ℝn), to weak Lp(ℝn) space for max{2n2n+1,nn+α}<p<1, if Ω satisfies the L1,α-Dini condition with any 0<α≤1.http://dx.doi.org/10.1155/2010/271905
collection DOAJ
language English
format Article
sources DOAJ
author Xiangxing Tao
Xiao Yu
Songyan Zhang
spellingShingle Xiangxing Tao
Xiao Yu
Songyan Zhang
Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces
Journal of Function Spaces and Applications
author_facet Xiangxing Tao
Xiao Yu
Songyan Zhang
author_sort Xiangxing Tao
title Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces
title_short Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces
title_full Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces
title_fullStr Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces
title_full_unstemmed Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces
title_sort marcinkiewicz integrals with variable kernels on hardy and weak hardy spaces
publisher Hindawi Limited
series Journal of Function Spaces and Applications
issn 0972-6802
publishDate 2010-01-01
description In this article, we consider the Marcinkiewicz integrals with variable kernels defined by μΩ(f)(x)=(∫0∞|∫|x−y|≤tΩ(x,x−y)|x−y|n−1f(y)dy|2dtt3)1/2, where Ω(x,z)∈L∞(ℝn)×Lq(Sn−1) for q > 1. We prove that the operator μΩ is bounded from Hardy space, Hp(ℝn), to Lp(ℝn) space; and is bounded from weak Hardy space, Hp,∞(ℝn), to weak Lp(ℝn) space for max{2n2n+1,nn+α}<p<1, if Ω satisfies the L1,α-Dini condition with any 0<α≤1.
url http://dx.doi.org/10.1155/2010/271905
work_keys_str_mv AT xiangxingtao marcinkiewiczintegralswithvariablekernelsonhardyandweakhardyspaces
AT xiaoyu marcinkiewiczintegralswithvariablekernelsonhardyandweakhardyspaces
AT songyanzhang marcinkiewiczintegralswithvariablekernelsonhardyandweakhardyspaces
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