International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2006 / Article

Open Access

Volume 2006 |Article ID 031705 | https://doi.org/10.1155/IJMMS/2006/31705

Shanzhen Lu, Huixia Mo, "Boundedness of higher-order Marcinkiewicz-Type integrals", International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 031705, 21 pages, 2006. https://doi.org/10.1155/IJMMS/2006/31705

Boundedness of higher-order Marcinkiewicz-Type integrals

Received11 Apr 2005
Revised20 Nov 2005
Accepted05 Dec 2005
Published24 May 2006

Abstract

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.

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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.


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