Boundedness of Littlewood-Paley Operators Associated with Gauss Measures
Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space (𝒳,d,μ)ρ, which means that the set 𝒳 is endowed with a metric d and a locally doubling regular Borel measure μ satisfying doubling and reverse dou...
Main Authors: | , |
---|---|
Format: | Article |
Language: | English |
Published: |
SpringerOpen
2010-01-01
|
Series: | Journal of Inequalities and Applications |
Online Access: | http://dx.doi.org/10.1155/2010/643948 |
id |
doaj-d9dbe94d66654ed8a3323e8aaade0be8 |
---|---|
record_format |
Article |
spelling |
doaj-d9dbe94d66654ed8a3323e8aaade0be82020-11-25T00:13:10ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2010-01-01201010.1155/2010/643948Boundedness of Littlewood-Paley Operators Associated with Gauss MeasuresLiguang LiuDachun YangModeled on the Gauss measure, the authors introduce the locally doubling measure metric space (𝒳,d,μ)ρ, which means that the set 𝒳 is endowed with a metric d and a locally doubling regular Borel measure μ satisfying doubling and reverse doubling conditions on admissible balls defined via the metric d and certain admissible function ρ. The authors then construct an approximation of the identity on (𝒳,d,μ)ρ, which further induces a Calderón reproducing formula in Lp(𝒳) for p∈(1,∞). Using this Calderón reproducing formula and a locally variant of the vector-valued singular integral theory, the authors characterize the space Lp(𝒳) for p∈(1,∞) in terms of the Littlewood-Paley g-function which is defined via the constructed approximation of the identity. Moreover, the authors also establish the Fefferman-Stein vector-valued maximal inequality for the local Hardy-Littlewood maximal function on (𝒳,d,μ)ρ. All results in this paper can apply to various settings including the Gauss measure metric spaces with certain admissible functions related to the Ornstein-Uhlenbeck operator, and Euclidean spaces and nilpotent Lie groups of polynomial growth with certain admissible functions related to Schrödinger operators. http://dx.doi.org/10.1155/2010/643948 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Liguang Liu Dachun Yang |
spellingShingle |
Liguang Liu Dachun Yang Boundedness of Littlewood-Paley Operators Associated with Gauss Measures Journal of Inequalities and Applications |
author_facet |
Liguang Liu Dachun Yang |
author_sort |
Liguang Liu |
title |
Boundedness of Littlewood-Paley Operators Associated with Gauss Measures |
title_short |
Boundedness of Littlewood-Paley Operators Associated with Gauss Measures |
title_full |
Boundedness of Littlewood-Paley Operators Associated with Gauss Measures |
title_fullStr |
Boundedness of Littlewood-Paley Operators Associated with Gauss Measures |
title_full_unstemmed |
Boundedness of Littlewood-Paley Operators Associated with Gauss Measures |
title_sort |
boundedness of littlewood-paley operators associated with gauss measures |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1025-5834 1029-242X |
publishDate |
2010-01-01 |
description |
Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space (𝒳,d,μ)ρ, which means that the set 𝒳 is endowed with a metric d and a locally doubling regular Borel measure μ satisfying doubling and reverse doubling conditions on admissible balls defined via the metric d and certain admissible function ρ. The authors then construct an approximation of the identity on (𝒳,d,μ)ρ, which further induces a Calderón reproducing formula in Lp(𝒳) for p∈(1,∞). Using this Calderón reproducing formula and a locally variant of the vector-valued singular integral theory, the authors characterize the space Lp(𝒳) for p∈(1,∞) in terms of the Littlewood-Paley g-function which is defined via the constructed approximation of the identity. Moreover, the authors also establish the Fefferman-Stein vector-valued maximal inequality for the local Hardy-Littlewood maximal function on (𝒳,d,μ)ρ. All results in this paper can apply to various settings including the Gauss measure metric spaces with certain admissible functions related to the Ornstein-Uhlenbeck operator, and Euclidean spaces and nilpotent Lie groups of polynomial growth with certain admissible functions related to Schrödinger operators. |
url |
http://dx.doi.org/10.1155/2010/643948 |
work_keys_str_mv |
AT liguangliu boundednessoflittlewoodpaleyoperatorsassociatedwithgaussmeasures AT dachunyang boundednessoflittlewoodpaleyoperatorsassociatedwithgaussmeasures |
_version_ |
1725396074023092224 |