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International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 31705, 21 pages

Boundedness of higher-order Marcinkiewicz-Type integrals

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received 11 April 2005; Revised 20 November 2005; Accepted 5 December 2005

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let A be a function with derivatives of order m and DγAΛ˙β(0<β<1,|γ|=m). The authors in the paper proved that if ΩLs(Sn1) (sn/(nβ)) is homogeneous of degree zero and satisfies a vanishing condition, then both the higher-order Marcinkiewicz-type integral μΩA and its variation μ˜ΩA are bounded from Lp(n) to Lq(n) and from L1(n) to Ln/(nβ),(n), where 1<p<n/β and 1/q=1/pβ/n. Furthermore, if Ω satisfies some kind of Ls-Dini condition, then both μΩA and μ˜ΩA are bounded on Hardy spaces, and μΩA is also bounded from Lp(n) to certain Triebel-Lizorkin space.