Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures
Let μ be a positive Radon measure on ℝd which may be nondoubling. The only condition that μ satisfies is μ(B(x,r))≤C0rn for all x∈ℝd, r>0, and some fixed constant C0. In this paper, we introduce the operator g&am...
Main Authors: | , |
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Format: | Article |
Language: | English |
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SpringerOpen
2009-01-01
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Series: | Journal of Inequalities and Applications |
Online Access: | http://dx.doi.org/10.1155/2009/175230 |
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author | Qingying Xue Juyang Zhang |
author_facet | Qingying Xue Juyang Zhang |
author_sort | Qingying Xue |
collection | DOAJ |
description | Let μ be a positive Radon measure on ℝd which may be nondoubling. The only condition that μ satisfies is μ(B(x,r))≤C0rn for all x∈ℝd, r>0, and some fixed constant C0. In this paper, we introduce the operator gλ,μ∗ related to such a measure and assume it is bounded on L2(μ). We then establish its boundedness, respectively, from the Lebesgue space L1(μ) to the weak Lebesgue space L1,∞(μ), from the Hardy space H1(μ) to L1(μ) and from the Lesesgue space L∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of gλ,μ∗ in the Lebesgue space Lp(μ) with p∈(1,∞). |
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id | doaj.art-ad81c6e72989440288ab544690fb7146 |
institution | Directory Open Access Journal |
issn | 1025-5834 1029-242X |
language | English |
last_indexed | 2024-04-12T11:42:14Z |
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series | Journal of Inequalities and Applications |
spelling | doaj.art-ad81c6e72989440288ab544690fb71462022-12-22T03:34:36ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2009-01-01200910.1155/2009/175230Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling MeasuresQingying XueJuyang ZhangLet μ be a positive Radon measure on ℝd which may be nondoubling. The only condition that μ satisfies is μ(B(x,r))≤C0rn for all x∈ℝd, r>0, and some fixed constant C0. In this paper, we introduce the operator gλ,μ∗ related to such a measure and assume it is bounded on L2(μ). We then establish its boundedness, respectively, from the Lebesgue space L1(μ) to the weak Lebesgue space L1,∞(μ), from the Hardy space H1(μ) to L1(μ) and from the Lesesgue space L∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of gλ,μ∗ in the Lebesgue space Lp(μ) with p∈(1,∞).http://dx.doi.org/10.1155/2009/175230 |
spellingShingle | Qingying Xue Juyang Zhang Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures Journal of Inequalities and Applications |
title | Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures |
title_full | Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures |
title_fullStr | Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures |
title_full_unstemmed | Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures |
title_short | Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures |
title_sort | endpoint estimates for a class of littlewood paley operators with nondoubling measures |
url | http://dx.doi.org/10.1155/2009/175230 |
work_keys_str_mv | AT qingyingxue endpointestimatesforaclassoflittlewoodpaleyoperatorswithnondoublingmeasures AT juyangzhang endpointestimatesforaclassoflittlewoodpaleyoperatorswithnondoublingmeasures |