BOUNDEDNESS OF GENERALIZED HIGHER COMMUTATORS OF MARCINKIEWICZ INTEGRALS

doi:10.1016/S0252-9602(07)60083-6

Abstract

Abstract:

Let $\vec{b}=(b_{1},\cdots ,b_{m})$ be a finite family of locally
integrable functions. Then, we introduce generalized higher commutator of Marcinkiwicz integral as follows:

μ_{Ω}^{\vec{b}} (f) (x) = (\int_{0}^{\infty} | F_{Ω, t}^{\vec{b}} (f) (x) |^{2} \frac{d t}{t})^{1 / 2},

$\mu_{\Omega}^{\vec{b}}(f)(x)=\Bigl(\int_{0}^{\infty} |F_{\Omega,t}^{\vec{b}}(f)(x)|^{2}\frac{{\rm d}t}{t}\Bigr)^{1/2},$ where

F_{Ω, t}^{\vec{b}} (f) (x) = \frac{1}{t} \int_{| x - y | \leq t} \frac{Ω (x - y)}{| x - y |^{n - 1}} \prod_{j = 1}^{m} (b_{j} (x) - b_{j} (y)) f (y) d y .

$F_{\Omega,t}^{\vec{b}}(f)(x)=\frac{1}{t}\int_{|x-y|\leq t} \frac{\Omega(x-y)}{|x-y|^{n-1}}\prod\limits_{j=1}^{m}(b_{j}(x)-b_{j}(y))f(y){\rm d}y.$
When

b_{j} \in {\dot{Λ}}_{β_{j}}, 1 \leq j \leq m,

$b_{j}\in\dot{\Lambda}_{\beta_{j}}, 1\leq j\leq m,$

0 < β_{j} < 1, \sum_{j = 1}^{m} β_{j} = \betais h o m o g e n e o u s o f d e g r e e z e r o a n d s a t i s f i e s t h e c a n c e l a t i o n c o n d i t i o n, w e p r o v e t h a t

$0<\beta_{j}<1, \sum\limits_{j=1}^{m}\beta_{j}=\betais homogeneous of degree zero and satisfies the cancelation condition, we prove that$ \mu_{\Omega}^{\vec{b}}

i s b o u n d e d f r o m

$is bounded from$ L^{p}({\Bbb R}^{n})

t o

$to$ L^{s}({\Bbb R}^{n}),

w h e r e

$where$ 1Moreover, if

Ω

$\Omega$ also satisfies some L^q-Dini condition, then

μ_{Ω}^{\vec{b}}

$\mu_{\Omega}^{\vec{b}}$ is bounded from

L^{p} (R^{n})

$L^{p}({\Bbb R}^{n})$ to

{\dot{F}}_{p}^{β, \infty} (R^{n})

$\dot{F}^{\beta,\infty}_{p}({\Bbb R}^{n})$ and on certain Hardy spaces.
The article extends some known results.

Key words: Generalized higher commutator of Marcinkiewicz integral, Hardy space, Lipschitz space

CLC Number:

42B20

Mo Huixia; Lu Shanzhen. BOUNDEDNESS OF GENERALIZED HIGHER COMMUTATORS OF MARCINKIEWICZ INTEGRALS[J].Acta mathematica scientia,Series B, 2007, 27(4): 852-866.

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