On weighted inequalities for parametric Marcinkiewicz integrals

• Main Author: Al-Qassem HM
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2006/91541
• ISSN: 1025-5834; 1029-242X

<p/> <p>We establish a weighted <inline-formula><graphic file="1029-242X-2006-91541-i1.gif"/></inline-formula> boundedness of a parametric Marcinkiewicz integral operator <inline-formula><graphic file="1029-242X-2006-91541-i2.gif"/></inline-formula> if <inline-formula><graphic file="1029-242X-2006-91541-i3.gif"/></inline-formula> is allowed to be in the block space <inline-formula><graphic file="1029-242X-2006-91541-i4.gif"/></inline-formula> for some <inline-formula><graphic file="1029-242X-2006-91541-i5.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2006-91541-i6.gif"/></inline-formula> satisfies a mild integrability condition. We apply this conclusion to obtain the weighted <inline-formula><graphic file="1029-242X-2006-91541-i7.gif"/></inline-formula> boundedness for a class of the parametric Marcinkiewicz integral operators <inline-formula><graphic file="1029-242X-2006-91541-i8.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2006-91541-i9.gif"/></inline-formula> related to the Littlewood-Paley <inline-formula><graphic file="1029-242X-2006-91541-i10.gif"/></inline-formula>-function and the area integral <inline-formula><graphic file="1029-242X-2006-91541-i11.gif"/></inline-formula>, respectively. It is known that the condition <inline-formula><graphic file="1029-242X-2006-91541-i12.gif"/></inline-formula> is optimal for the <inline-formula><graphic file="1029-242X-2006-91541-i13.gif"/></inline-formula> boundedness of <inline-formula><graphic file="1029-242X-2006-91541-i14.gif"/></inline-formula>.</p>