Boundedness of higher-order Marcinkiewicz-Type integrals
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 ar...
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| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2006-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS/2006/31705 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |