Volume 28, Issue 3
BMO Estimates for Multilinar Fractional Integrals
10.3969/j.issn.1672-4070.2012.03.002

Anal. Theory Appl., 28 (2012), pp. 224-231

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• Abstract

In this paper, the authors prove that the multilinear fractional integral operator$T_{\Omega,\alpha}^{A1,A2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A1,A2}$ W,a with rough kernel are both bounded from$L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−a),\infty$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A1,A2}( f )(x) =\int_{R^n}\frac{{R_{m_1}}(A_1;x,y){R_{m_2}}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2_2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A1,A2}(f)(x)= \sup_{r & gt 0}\frac{1}{r^n−\alpha +m_1+m_2−2}\int_{|x−y| \< r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0 < \alpha < n$, $\Omega \in L^s(S^n−1)$ $(s \geq 1)$ is a homogeneous function of degree zero in $R^n$, $A_i$ is a function defined on $R^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor seriesof $A_i$ at $x$ about $y$. More precisely, $R_{m_i}$ (A_i;x,y) = A_i(x)− \sum_{|\gamma|< m_i}\frac{1}{\gamma!}D^\gammaA_i(y)(x−y)^r,$where$D^\gamma (Ai) \in BMO(R^n)$for$|\gamma | = m_i−1(m_i > 1)$,$i = 1,2\$.

• History

Published online: 2012-10

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