Boundedness of Littlewood-Paley Operators Associated with Gauss Measures

• Main Authors: Liu Liguang, Yang Dachun
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2010/643948

<p>Abstract</p> <p>Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space <inline-formula> <graphic file="1029-242X-2010-643948-i1.gif"/></inline-formula>, which means that the set <inline-formula> <graphic file="1029-242X-2010-643948-i2.gif"/></inline-formula> is endowed with a metric <inline-formula> <graphic file="1029-242X-2010-643948-i3.gif"/></inline-formula> and a locally doubling regular Borel measure <inline-formula> <graphic file="1029-242X-2010-643948-i4.gif"/></inline-formula> satisfying doubling and reverse doubling conditions on admissible balls defined via the metric <inline-formula> <graphic file="1029-242X-2010-643948-i5.gif"/></inline-formula> and certain admissible function <inline-formula> <graphic file="1029-242X-2010-643948-i6.gif"/></inline-formula>. The authors then construct an approximation of the identity on <inline-formula> <graphic file="1029-242X-2010-643948-i7.gif"/></inline-formula>, which further induces a Calder&#243;n reproducing formula in <inline-formula> <graphic file="1029-242X-2010-643948-i8.gif"/></inline-formula> for <inline-formula> <graphic file="1029-242X-2010-643948-i9.gif"/></inline-formula>. Using this Calder&#243;n reproducing formula and a locally variant of the vector-valued singular integral theory, the authors characterize the space <inline-formula> <graphic file="1029-242X-2010-643948-i10.gif"/></inline-formula> for <inline-formula> <graphic file="1029-242X-2010-643948-i11.gif"/></inline-formula> in terms of the Littlewood-Paley <inline-formula> <graphic file="1029-242X-2010-643948-i12.gif"/></inline-formula>-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 <inline-formula> <graphic file="1029-242X-2010-643948-i13.gif"/></inline-formula>. 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 Schr&#246;dinger operators.</p>