International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 31705, 21 pages
doi:10.1155/IJMMS/2006/31705
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 Shanzhen Lu and Huixia Mo. 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.
Abstract
Let A
be a function with derivatives of order m
and
DγA∈Λ˙β(0<β<1,|γ|=m).
The authors in the paper proved that if Ω∈Ls(Sn−1) (s≥n/(n−β))
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(ℝn)
to Lq(ℝn)
and from
L1(ℝn)
to
Ln/(n−β),∞(ℝn), where 1<p<n/β
and
1/q=1/p−β/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(ℝn)
to
certain Triebel-Lizorkin space.