多线性分数次积分算子交换子的有界性 Boundedness of multilinear fractional integral commutators期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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多线性分数次积分算子交换子的有界性

作者姓名：

Wang Lei Pan Ting Dept. of Math. Zhejiang Univ. Hangzhou China. Univ. of International Relation Hangzhou China.

作者单位：

Wang Lei Pan Ting1 Dept. of Math.,Zhejiang Univ.,Hangzhou 310028,China. 2 Univ. of International Relation,Hangzhou 310015,China.

摘要：

Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(…

关键词：

积分算子卢森堡标准多重线性转接器向量

Boundedness of multilinear fractional integral commutators

Wang Lei Pan Ting Dept. of Math.,Zhejiang Univ.,Hangzhou ,China. Univ. of International Relation,Hangzhou ,China..Boundedness of multilinear fractional integral commutators[J].Applied Mathematics A Journal of Chinese Universities,2004,19(2):212-222.

Authors:

Wang Lei Pan Ting

Institution:

Dept.of Math.,Zhejiang Univ.,Hangzhou 310028,China.;Univ.of International Relation,Hangzhou 310015,China.

Abstract:

By introducing a kind of maximal operator of fractional order associated with the mean Luxemburg norm and using the technique of Sharp function, multilinear commutators of fractional integral operator with vector symbol b = (b ₁,...b _m)defined by

$I_\alpha ^b f(x) = \int_R {\left {\prod\limits_{j = 1}^m {(b_j (x) - b_j (y))} } \right]} \frac{1}{{|x - y|^{n - \alpha } }}f(y)\user2{d}y$

are considered. The following priori estimates are proved. For 1<p<∞, there exists a constant c such that

$\parallel I_\alpha ^b f\parallel L^p (R^n ) \leqslant c\parallel b \parallel \parallel M_{L(logL)} \tfrac{l}{r},_\alpha (f)\parallel L^p (R_n ),$

$\parallel I_\alpha ^b f\parallel L^q (R^n ) \leqslant c\parallel b \parallel \parallel f\parallel L^p (R_n ),$

where 1<p<n/α,1/q=1/p−α/n,0<α<n,and

$\begin{gathered} \mathop {\sup }\limits_{t > 0} \frac{1}{{\Phi (\tfrac{1}{t})}} \left| { \left\{ {y \in R^n : \left| { I_a^b f\left( y \right) } \right| > t} \right\}} \right|\tfrac{1}{q} \leqslant \hfill \\ c \mathop {\sup }\limits_{t > 0} \frac{1}{{\Phi (\tfrac{1}{t})}}\left| {\left\{ {y \in R^n :M_{L(logL)} \tfrac{1}{r}_{,a} \left( { \left\| { b } \right\| f} \right)\left( y \right) > t} \right\}} \right|\tfrac{1}{q}, \hfill \\ \end{gathered}$

where $\parallel b\parallel = \prod\nolimits_{j - 1}^m {\parallel b_j \parallel } _{Osc_{expL^{rj} } } ,\Phi (1 + log^ + t)^{\tfrac{1}{r}} ,\frac{1}{r} = \frac{1}{{r_1 }} + \cdots + \frac{1}{{r_m }},$ $M_{L(logL)} \tfrac{1}{{r,\alpha }}$ is an Orlicz type maximal operator.

Keywords:

fractional integral operator commutator sharp function Young function Luxemburg function

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