Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures

• Main Authors: Xue Qingying, Zhang Juyang
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2009/175230

<p/> <p>Let <inline-formula> <graphic file="1029-242X-2009-175230-i1.gif"/></inline-formula> be a positive Radon measure on <inline-formula> <graphic file="1029-242X-2009-175230-i2.gif"/></inline-formula> which may be nondoubling. The only condition that <inline-formula> <graphic file="1029-242X-2009-175230-i3.gif"/></inline-formula> satisfies is <inline-formula> <graphic file="1029-242X-2009-175230-i4.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2009-175230-i5.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2009-175230-i6.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2009-175230-i7.gif"/></inline-formula>, and some fixed constant <inline-formula> <graphic file="1029-242X-2009-175230-i8.gif"/></inline-formula>. In this paper, we introduce the operator <inline-formula> <graphic file="1029-242X-2009-175230-i9.gif"/></inline-formula> related to such a measure and assume it is bounded on <inline-formula> <graphic file="1029-242X-2009-175230-i10.gif"/></inline-formula>. We then establish its boundedness, respectively, from the Lebesgue space <inline-formula> <graphic file="1029-242X-2009-175230-i11.gif"/></inline-formula> to the weak Lebesgue space <inline-formula> <graphic file="1029-242X-2009-175230-i12.gif"/></inline-formula>, from the Hardy space <inline-formula> <graphic file="1029-242X-2009-175230-i13.gif"/></inline-formula> to <inline-formula> <graphic file="1029-242X-2009-175230-i14.gif"/></inline-formula> and from the Lesesgue space <inline-formula> <graphic file="1029-242X-2009-175230-i15.gif"/></inline-formula> to the space <inline-formula> <graphic file="1029-242X-2009-175230-i16.gif"/></inline-formula>. As a corollary, we obtain the boundedness of <inline-formula> <graphic file="1029-242X-2009-175230-i17.gif"/></inline-formula> in the Lebesgue space <inline-formula> <graphic file="1029-242X-2009-175230-i18.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2009-175230-i19.gif"/></inline-formula>.</p>