Boundedness of higher-order Marcinkiewicz-Type integrals

Shanzhen Lu and Huixia Mo

International Journal of Mathematics and Mathematical Sciences, 2006, vol. 2006, 1-21

Abstract:

Let A be a function with derivatives of order m and D γ A ∈ Λ ˙ β ( 0 < β < 1 , | γ | = m ) . The authors in the paper proved that if Ω ∈ L s ( S n − 1 )  ( s ≥ n / ( n − β ) ) is homogeneous of degree zero and satisfies a vanishing condition, then both the higher-order Marcinkiewicz-type integral μ Ω A and its variation μ ˜ Ω A are bounded from L p ( ℠n ) to L q ( ℠n ) and from L 1 ( ℠n ) to L n / ( n − β ) , ∞ ( ℠n ) , where 1 < p < n / β and 1 / q = 1 / p − β / n . Furthermore, if Ω satisfies some kind of L s -Dini condition, then both μ Ω A and μ ˜ Ω A are bounded on Hardy spaces, and μ Ω A is also bounded from L p ( ℠n ) to certain Triebel-Lizorkin space.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:031705

DOI: 10.1155/IJMMS/2006/31705

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